% Complex Analytic and Differential Geometry, Chapter I
% J.-P. Demailly, Universit\'e de Grenoble I, Saint Martin d'H\`eres, France

%% Personal macros
\def\bigger{1200} \def\smaller{1000} \let\textsize=\bigger

\def\swapleft#1#2{\hbox{\rlap{#2}\kern\parindent\kern-#1}}

\ifx\textsize\smaller
\magnification=\textsize
\vsize=24.48cm \hsize=15.8cm
\parskip=3pt plus 2pt minus 1pt
\parindent=6.24mm
\else
\magnification=\textsize
\vsize=20.4cm \hsize=13.5cm
\parskip=4pt plus 2pt minus 1pt
\parindent=5.2mm
\fi

\font\booktitlefont = cmr10 at 35.832pt
\font\headfont = cmbx10 at 20.736pt
\font\tenrmmfour = cmr10 at 20.736pt
\font\tenrmmthree = cmr10 at 17.28pt

\font\fourteenbf = cmbx10 at 14.4pt
\font\twelvebf = cmbx10 at 12pt

\font\fourteenbsy=cmbsy10 at 14.4pt
\font\twelvebsy=cmbsy10 at 12pt
\font\tenbsy=cmbsy10
\font\eightbsy=cmbsy8
\font\sevenbsy=cmbsy7
\font\sixbsy=cmbsy6
\font\fivebsy=cmbsy5

%\font\tensans = cmss10
%\font\eightsans = cmss8
%\font\sevensans = cmss7
%\font\sixsans = cmss6
%\font\fivesans = cmss5
%\newfam\sansfam
%\textfont\sansfam=\tensans
%\scriptfont\sansfam=\sevensans
%\scriptscriptfont\sansfam=\fivesans
%\def\sans{\fam\sansfam\tensans}

%\font\tenCal=mrsfs10 at 10pt
%\font\eightCal=mrsfs10 at 8pt
%\font\sevenCal=mrsfs10 at 7pt
%\font\sixCal=mrsfs10 at 6pt
%\font\fiveCal=mrsfs10 at 5pt

\font\tenCal=caljpd10 at 10pt
\font\eightCal=caljpd10 at 8pt
\font\sevenCal=caljpd10 at 7pt
\font\sixCal=caljpd10 at 6pt
\font\fiveCal=caljpd10 at 5pt

%\font\tenCal=ES at 10pt
%\font\eightCal=ES at 8pt
%\font\sevenCal=ES at 7pt
%\font\sixCal=ES at 6pt
%\font\fiveCal=ES at 5pt

\newfam\Calfam
  \textfont\Calfam=\tenCal
  \scriptfont\Calfam=\sevenCal
  \scriptscriptfont\Calfam=\fiveCal
\def\Cal{\fam\Calfam\tenCal}

%% AMS stuff

\font\tenmsa = msam10
\font\sevenmsa = msam7
\font\fivemsa = msam5
\newfam\msafam
\textfont\msafam=\tenmsa
\scriptfont\msafam=\sevenmsa
\scriptscriptfont\msafam=\fivemsa

\font\tenmsb = msbm10
\font\sevenmsb = msbm7
\font\fivemsb = msbm5
\newfam\msbfam
\textfont\msbfam=\tenmsb
\scriptfont\msbfam=\sevenmsb
\scriptscriptfont\msbfam=\fivemsb

\font\teneuf = eufm10
\font\seveneuf = eufm7
\font\fiveeuf = eufm5
\newfam\euffam
\textfont\euffam=\teneuf
\scriptfont\euffam=\seveneuf
\scriptscriptfont\euffam=\fiveeuf

\catcode`\@=11
\def\frak#1{{\fam\euffam#1}}
\def\Bbb#1{{\fam\msbfam#1}}
\def\hexnumber@#1{\ifcase#1 0\or1\or2\or3\or4\or5\or6\or7\or8\or9\or
 A\or B\or C\or D\or E\or F\fi}
\edef\msa@{\hexnumber@\msafam}
\edef\msb@{\hexnumber@\msbfam}
\mathchardef\boxdot="2\msa@00
\mathchardef\boxplus="2\msa@01
\mathchardef\boxtimes="2\msa@02
\mathchardef\square="0\msa@03
\mathchardef\blacksquare="0\msa@04
\mathchardef\centerdot="2\msa@05
\mathchardef\lozenge="0\msa@06
\mathchardef\blacklozenge="0\msa@07
\mathchardef\circlearrowright="3\msa@08
\mathchardef\circlearrowleft="3\msa@09
\mathchardef\rightleftharpoons="3\msa@0A
\mathchardef\leftrightharpoons="3\msa@0B
\mathchardef\boxminus="2\msa@0C
\mathchardef\Vdash="3\msa@0D
\mathchardef\Vvdash="3\msa@0E
\mathchardef\vDash="3\msa@0F
\mathchardef\twoheadrightarrow="3\msa@10
\mathchardef\twoheadleftarrow="3\msa@11
\mathchardef\leftleftarrows="3\msa@12
\mathchardef\rightrightarrows="3\msa@13
\mathchardef\upuparrows="3\msa@14
\mathchardef\downdownarrows="3\msa@15
\mathchardef\upharpoonright="3\msa@16
\let\restriction\upharpoonright
\mathchardef\downharpoonright="3\msa@17
\mathchardef\upharpoonleft="3\msa@18
\mathchardef\downharpoonleft="3\msa@19
\mathchardef\rightarrowtail="3\msa@1A
\mathchardef\leftarrowtail="3\msa@1B
\mathchardef\leftrightarrows="3\msa@1C
\mathchardef\rightleftarrows="3\msa@1D
\mathchardef\Lsh="3\msa@1E
\mathchardef\Rsh="3\msa@1F
\mathchardef\rightsquigarrow="3\msa@20
\mathchardef\leftrightsquigarrow="3\msa@21
\mathchardef\looparrowleft="3\msa@22
\mathchardef\looparrowright="3\msa@23
\mathchardef\circeq="3\msa@24
\mathchardef\succsim="3\msa@25
\mathchardef\gtrsim="3\msa@26
\mathchardef\gtrapprox="3\msa@27
\mathchardef\multimap="3\msa@28
\mathchardef\therefore="3\msa@29
\mathchardef\because="3\msa@2A
\mathchardef\doteqdot="3\msa@2B
\let\Doteq\doteqdot
\mathchardef\triangleq="3\msa@2C
\mathchardef\precsim="3\msa@2D
\mathchardef\lesssim="3\msa@2E
\mathchardef\lessapprox="3\msa@2F
\mathchardef\eqslantless="3\msa@30
\mathchardef\eqslantgtr="3\msa@31
\mathchardef\curlyeqprec="3\msa@32
\mathchardef\curlyeqsucc="3\msa@33
\mathchardef\preccurlyeq="3\msa@34
\mathchardef\leqq="3\msa@35
\mathchardef\leqslant="3\msa@36 \let\le=\leqslant
\mathchardef\lessgtr="3\msa@37
\mathchardef\backprime="0\msa@38
\mathchardef\risingdotseq="3\msa@3A
\mathchardef\fallingdotseq="3\msa@3B
\mathchardef\succcurlyeq="3\msa@3C
\mathchardef\geqq="3\msa@3D
\mathchardef\geqslant="3\msa@3E  \let\ge=\geqslant
\mathchardef\gtrless="3\msa@3F
\mathchardef\sqsubset="3\msa@40
\mathchardef\sqsupset="3\msa@41
\mathchardef\vartriangleright="3\msa@42
\mathchardef\vartriangleleft ="3\msa@43
\mathchardef\trianglerighteq="3\msa@44
\mathchardef\trianglelefteq="3\msa@45
\mathchardef\bigstar="0\msa@46
\mathchardef\between="3\msa@47
\mathchardef\blacktriangledown="0\msa@48
\mathchardef\blacktriangleright="3\msa@49
\mathchardef\blacktriangleleft="3\msa@4A
\mathchardef\vartriangle="0\msa@4D
\mathchardef\blacktriangle="0\msa@4E
\mathchardef\triangledown="0\msa@4F
\mathchardef\eqcirc="3\msa@50
\mathchardef\lesseqgtr="3\msa@51
\mathchardef\gtreqless="3\msa@52
\mathchardef\lesseqqgtr="3\msa@53
\mathchardef\gtreqqless="3\msa@54
\mathchardef\Rrightarrow="3\msa@56
\mathchardef\Lleftarrow="3\msa@57
\mathchardef\veebar="2\msa@59
\mathchardef\barwedge="2\msa@5A
\mathchardef\doublebarwedge="2\msa@5B
\mathchardef\angle="0\msa@5C
\mathchardef\measuredangle="0\msa@5D
\mathchardef\sphericalangle="0\msa@5E
\mathchardef\varpropto="3\msa@5F
\mathchardef\smallsmile="3\msa@60
\mathchardef\smallfrown="3\msa@61
\mathchardef\Subset="3\msa@62
\mathchardef\Supset="3\msa@63
\mathchardef\Cup="2\msa@64
\let\doublecup\Cup
\mathchardef\Cap="2\msa@65
\let\doublecap\Cap
\mathchardef\curlywedge="2\msa@66
\mathchardef\curlyvee="2\msa@67
\mathchardef\leftthreetimes="2\msa@68
\mathchardef\rightthreetimes="2\msa@69
\mathchardef\subseteqq="3\msa@6A
\mathchardef\supseteqq="3\msa@6B
\mathchardef\bumpeq="3\msa@6C
\mathchardef\Bumpeq="3\msa@6D
\mathchardef\lll="3\msa@6E
\let\llless\lll
\mathchardef\ggg="3\msa@6F
\let\gggtr\ggg
\mathchardef\circledS="0\msa@73
\mathchardef\pitchfork="3\msa@74
\mathchardef\dotplus="2\msa@75
\mathchardef\backsim="3\msa@76
\mathchardef\backsimeq="3\msa@77
\mathchardef\complement="0\msa@7B
\mathchardef\intercal="2\msa@7C
\mathchardef\circledcirc="2\msa@7D
\mathchardef\circledast="2\msa@7E
\mathchardef\circleddash="2\msa@7F
\def\ulcorner{\delimiter"4\msa@70\msa@70 }
\def\urcorner{\delimiter"5\msa@71\msa@71 }
\def\llcorner{\delimiter"4\msa@78\msa@78 }
\def\lrcorner{\delimiter"5\msa@79\msa@79 }
\def\yen{{\mathhexbox@\msa@55 }}
\def\checkmark{{\mathhexbox@\msa@58 }}
\def\circledR{{\mathhexbox@\msa@72 }}
\def\maltese{{\mathhexbox@\msa@7A }}
\mathchardef\lvertneqq="3\msb@00
\mathchardef\gvertneqq="3\msb@01
\mathchardef\nleq="3\msb@02
\mathchardef\ngeq="3\msb@03
\mathchardef\nless="3\msb@04
\mathchardef\ngtr="3\msb@05
\mathchardef\nprec="3\msb@06
\mathchardef\nsucc="3\msb@07
\mathchardef\lneqq="3\msb@08
\mathchardef\gneqq="3\msb@09
\mathchardef\nleqslant="3\msb@0A
\mathchardef\ngeqslant="3\msb@0B
\mathchardef\lneq="3\msb@0C
\mathchardef\gneq="3\msb@0D
\mathchardef\npreceq="3\msb@0E
\mathchardef\nsucceq="3\msb@0F
\mathchardef\precnsim="3\msb@10
\mathchardef\succnsim="3\msb@11
\mathchardef\lnsim="3\msb@12
\mathchardef\gnsim="3\msb@13
\mathchardef\nleqq="3\msb@14
\mathchardef\ngeqq="3\msb@15
\mathchardef\precneqq="3\msb@16
\mathchardef\succneqq="3\msb@17
\mathchardef\precnapprox="3\msb@18
\mathchardef\succnapprox="3\msb@19
\mathchardef\lnapprox="3\msb@1A
\mathchardef\gnapprox="3\msb@1B
\mathchardef\nsim="3\msb@1C
\mathchardef\napprox="3\msb@1D
\mathchardef\ncong="3\msb@1D
\def\napprox{\not\approx}
\mathchardef\varsubsetneq="3\msb@20
\mathchardef\varsupsetneq="3\msb@21
\mathchardef\nsubseteqq="3\msb@22
\mathchardef\nsupseteqq="3\msb@23
\mathchardef\subsetneqq="3\msb@24
\mathchardef\supsetneqq="3\msb@25
\mathchardef\varsubsetneqq="3\msb@26
\mathchardef\varsupsetneqq="3\msb@27
\mathchardef\subsetneq="3\msb@28
\mathchardef\supsetneq="3\msb@29
\mathchardef\nsubseteq="3\msb@2A
\mathchardef\nsupseteq="3\msb@2B
\mathchardef\nparallel="3\msb@2C
\mathchardef\nmid="3\msb@2D
\mathchardef\nshortmid="3\msb@2E
\mathchardef\nshortparallel="3\msb@2F
\mathchardef\nvdash="3\msb@30
\mathchardef\nVdash="3\msb@31
\mathchardef\nvDash="3\msb@32
\mathchardef\nVDash="3\msb@33
\mathchardef\ntrianglerighteq="3\msb@34
\mathchardef\ntrianglelefteq="3\msb@35
\mathchardef\ntriangleleft="3\msb@36
\mathchardef\ntriangleright="3\msb@37
\mathchardef\nleftarrow="3\msb@38
\mathchardef\nrightarrow="3\msb@39
\mathchardef\nLeftarrow="3\msb@3A
\mathchardef\nRightarrow="3\msb@3B
\mathchardef\nLeftrightarrow="3\msb@3C
\mathchardef\nleftrightarrow="3\msb@3D
\mathchardef\divideontimes="2\msb@3E
\mathchardef\varnothing="0\msb@3F
\mathchardef\nexists="0\msb@40
\mathchardef\mho="0\msb@66
\mathchardef\eth="0\msb@67
\mathchardef\eqsim="3\msb@68
\mathchardef\beth="0\msb@69
\mathchardef\gimel="0\msb@6A
\mathchardef\daleth="0\msb@6B
\mathchardef\lessdot="3\msb@6C
\mathchardef\gtrdot="3\msb@6D
\mathchardef\ltimes="2\msb@6E
\mathchardef\rtimes="2\msb@6F
\mathchardef\shortmid="3\msb@70
\mathchardef\shortparallel="3\msb@71
\mathchardef\smallsetminus="2\msb@72
\mathchardef\thicksim="3\msb@73
\mathchardef\thickapprox="3\msb@74
\mathchardef\approxeq="3\msb@75
\mathchardef\succapprox="3\msb@76
\mathchardef\precapprox="3\msb@77
\mathchardef\curvearrowleft="3\msb@78
\mathchardef\curvearrowright="3\msb@79
\mathchardef\digamma="0\msb@7A
\mathchardef\varkappa="0\msb@7B
\mathchardef\hslash="0\msb@7D
\mathchardef\hbar="0\msb@7E
\mathchardef\backepsilon="3\msb@7F
\catcode`\@=12

%% Private stuff

\def\ii{{\rm i}}
\def\bu{{\scriptstyle\bullet}}
\let\wh=\widehat
\let\wt=\widetilde
\let\ol=\overline
\let\ssm=\smallsetminus
\def\soul{{\raise-4.6pt\hbox{$-$}\kern-7pt}}
\def\compact{\subset\!\subset}
\def\ort{\hbox{~\vrule width4.0pt height0.4pt depth0pt
                \vrule width0.4pt height6.0pt depth0pt~$\,$}}

\def\lhra{\lhook\joinrel\longrightarrow}
\def\srelbar{\mathrel{\smash{\hbox{\rm-}}}}
\def\merto{\srelbar\,\srelbar\,\rightarrow}
\def\larex#1 {\hbox to #1 pt{\rightarrowfill}}
\def\mapstex#1 {\hbox to #1 pt{$\mapstochar$\rightarrowfill}}

\def\buildo#1\over#2{\mathrel{\mathop{\null#2}\limits^{#1}}}
\def\buildu#1\under#2{\mathrel{\mathop{\null#2}\limits_{#1}}}

\catcode`@=11    
\def\cmalign#1{\null\,\vcenter{\normalbaselines\m@th
    \ialign{\hfil$##$&&$##$\hfil\crcr
      \mathstrut\crcr\noalign{\kern-\baselineskip}
      #1\crcr\mathstrut\crcr\noalign{\kern-\baselineskip}}}\,}
\catcode`@=12

\font\fourteenmib = cmmib10 at 14.4pt
\font\twelvemib = cmmib10 at 12pt
\font\tenmib = cmmib10
\font\sevenmib = cmmib7

\font\eightrm = cmr8
\font\eightbf = cmbx8
\font\eighti = cmmi8
\font\eightit = cmti8
\font\eightsy = cmsy8
\font\eighttt = cmtt8
\font\eightex = cmex8
\font\eightmib = cmmib8
\font\eightmsa = msam8
\font\eightmsb = msbm8
\font\eighteuf = eufm8

\font\sixrm = cmr6
\font\sixbf = cmbx6
\font\sixi = cmmi6
\font\sixit = cmti6
\font\sixsy = cmsy6
\font\sixtt = cmtt6
\font\sixex = cmex6
\font\sixmib = cmmib6
\font\sixmsa = msam6
\font\sixmsb = msbm6
\font\sixeuf = eufm6

\font\fiverm = cmr5
\font\fivebf = cmbx5
\font\fivei = cmmi5
\font\fiveit = cmti5
\font\fivesy = cmsy5
\font\fivett = cmtt5
\font\fiveex = cmex5
\font\fivemib = cmmib5
\font\fivemsa = msam5
\font\fivemsb = msbm5

\def\eightpoint{\def\rm{\fam0\eightrm}% redefine \rm
   % set up smaller fonts
   \textfont0=\eightrm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm
   \textfont1=\eighti  \scriptfont1=\sixi  \scriptscriptfont1=\fivei
   \textfont2=\eightsy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy
   \textfont3=\eightex \scriptfont3=\eightex\scriptscriptfont3=\eightex
   % redefine all other font-family commands
   \def\it{\fam\itfam\eightit}%
   \textfont\itfam=\eightit
   \def\bf{\fam\bffam\eightbf}%
   \textfont\bffam=\eightbf \scriptfont\bffam=\sixbf
   \scriptscriptfont\bffam=\fivebf
   \def\tt{\fam\ttfam\eighttt}%
   \textfont\ttfam=\eighttt
   % redefine AmS fonts if they exist
   \textfont\msafam=\eightmsa \scriptfont\msafam=\sixmsa
   \textfont\msbfam=\eightmsb \scriptfont\msbfam=\sixmsb
   \textfont\euffam=\eighteuf \scriptfont\euffam=\sixeuf
   \textfont\Calfam=\eightCal
   \scriptfont\Calfam=\sixCal
   % new \baselineskip and \strut
   \normalbaselineskip=9.6pt
   % establish those values
   \normalbaselines\rm}

\newskip\eightpointsurround
\eightpointsurround=6pt plus 3pt minus 2pt
\def\begineightpoint{\vskip\eightpointsurround \bgroup\eightpoint}
\def\endeightpoint{\vskip\eightpointsurround \egroup}

\def\fourteenpointbf{%
 \textfont0=\fourteenbf   \scriptfont0=\twelvebf   \scriptscriptfont0=\sevenbf
 \textfont1=\fourteenmib  \scriptfont1=\twelvemib \scriptscriptfont1=\sevenmib
 \textfont2=\fourteenbsy  \scriptfont2=\twelvebsy  \scriptscriptfont2=\sevenbsy
 \fourteenbf
 \baselineskip=14.4pt}

\def\twelvepointbf{%
 \textfont0=\twelvebf   \scriptfont0=\eightbf   \scriptscriptfont0=\sixbf
 \textfont1=\twelvemib  \scriptfont1=\eightmib \scriptscriptfont1=\sixmib
 \textfont2=\twelvebsy  \scriptfont2=\eightbsy  \scriptscriptfont2=\sixbsy
 \twelvebf
 \baselineskip=14.4pt}

\def\tenpointbf{%
 \textfont0=\tenbf   \scriptfont0=\sevenbf   \scriptscriptfont0=\fivebf
 \textfont1=\tenmib  \scriptfont1=\sevenmib  \scriptscriptfont1=\fivemib
 \textfont2=\tenbsy  \scriptfont2=\sevenbsy  \scriptscriptfont2=\fivebsy
 \tenbf
 \baselineskip=14.4pt}

\def\proof#1{\removelastskip\vskip4pt\noindent{\it #1.}}
\def\endproof{\vskip5pt plus 1pt minus 2pt}

\def\qed{\relax\ifmmode
            \hbox{$\square$}%
         \else
            {\unskip\nobreak\hfil
            \penalty50\hskip1em\null\nobreak\hfil\hbox{$\square$}%
            \parfillskip=0pt\finalhyphendemerits=0\endgraf}%
         \fi}
\def\squareatend{\rlap{\hbox{\strut\kern\hsize\llap{$\square$}}}}

\long\def\claim#1 #2\endclaim
{\removelastskip\vskip7pt plus 1pt minus 1pt\noindent{\bf#1.}
{\it\ignorespaces#2}\vskip\baselineskip}
\def\\{\hfil\break}
\def\Fbreak{\hfill\break}

%% Frequently used symbols
\def\bC{{\Bbb C}}
\def\bH{{\Bbb H}}
\def\bK{{\Bbb K}}
\def\bN{{\Bbb N}}
\def\bP{{\Bbb P}}
\def\bQ{{\Bbb Q}}
\def\bR{{\Bbb R}}
\def\bZ{{\Bbb Z}}
\def\bOne{{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l}
  {\rm 1\mskip-4.5mu l} {\rm 1\mskip-5mu l}}}

\def\cA{{\Cal A}}
\def\cB{{\Cal B}}
\def\cC{{\Cal C}}
\def\cD{{\Cal D}}
\def\cE{{\Cal E}}
\def\cF{{\Cal F}}
\def\cG{{\Cal G}}
\def\cH{{\Cal H}}
\def\cI{{\Cal I}}
\def\cJ{{\Cal J}}
\def\cK{{\Cal K}}
\def\cL{{\Cal L}}
\def\cM{{\Cal M}}
\def\cN{{\Cal N}}
\def\cO{{\Cal O}}
\def\cP{{\Cal P}}
\def\cQ{{\Cal Q}}
\def\cR{{\Cal R}}
\def\cS{{\Cal S}}
\def\cT{{\Cal T}}
\def\cU{{\Cal U}}
\def\cV{{\Cal V}}
\def\cW{{\Cal W}}
\def\cX{{\Cal X}}
\def\cY{{\Cal Y}}
\def\cZ{{\Cal Z}}

\special{header=/home/demailly/Documents/psinputs/mathdraw/grlib.ps}
\newdimen\figmm
\figmm=0.694444mm
\long\def\InsertFig#1 #2 #3 #4\EndFig{\par
\hbox{\hskip #1\figmm$\vbox to #2\figmm{\vfil\special{" 
0.694444 dup scale
#3}}#4$}}
\long\def\LabelTeX#1 #2 #3\ELTX{\rlap{\kern#1\figmm\raise#2\figmm\hbox{#3}}}

\def\ga{{\frak a}}
\def\gm{{\frak m}}

\def\stimes{\mathop{\kern0.7pt
\vrule height 0.4pt depth 0pt width 5pt\kern-5pt
\vrule height 5.4pt depth -5pt width 5pt\kern-5pt
\vrule height 5.4pt depth 0pt width 0.4pt\kern4.6pt
\vrule height 5.4pt depth 0pt width 0.4pt\kern-6.5pt
\raise0.3pt\hbox{$\times$}\kern-0.7pt}\nolimits}
\def\hotimes{\mathrel{\widehat\otimes}}

\catcode`@=11
\def\rightarrowfil{\m@th\mathord-\mkern-6mu%
  \cleaders\hbox{$\mkern-2mu\mathord-\mkern-2mu$}\hfill
  \mkern-6mu\mathord\rightarrow}
\catcode`@=12


\def\gge{{\scriptscriptstyle\ge}}
\def\Ll{\langle\!\langle}
\def\Gg{\rangle\!\rangle}

\def\joinrel{\mathrel{\mkern-3.5mu}}
\def\dlra{\rlap{$\longrightarrow$}\kern-1pt\longrightarrow}
\def\llra{\relbar\joinrel\relbar\joinrel\longrightarrow}

\def\hookdown{\hbox{$\big\downarrow$\kern-4.2pt
\raise6.3pt\hbox{$\scriptscriptstyle\cap$}}}
\def\begarrow{\smash{\kern-0.5pt\vrule height-1.6pt depth2pt width4pt}}

\def\Ker{\mathop{\rm Ker}\nolimits}
\def\Id{\mathop{\rm Id}\nolimits}
\def\Vol{\mathop{\rm Vol}\nolimits}
\def\rk{\mathop{\rm rk}\nolimits}
\def\Tr{\mathop{\rm Tr}\nolimits}
\def\deg{\mathop{\rm deg}\nolimits}
\def\Re{\mathop{\rm Re}\nolimits}
\def\Im{\mathop{\rm Im}\nolimits}
\def\mod{\mathop{\rm mod}\nolimits}
\def\codim{\mathop{\rm codim}\nolimits}
\def\Supp{\mathop{\rm Supp}\nolimits}
\def\Coker{\mathop{\rm Coker}\nolimits}
\def\Hom{\mathop{\rm Hom}\nolimits}
\def\End{\mathop{\rm End}\nolimits}
\def\Aut{\mathop{\rm Aut}\nolimits}
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%% Start book

$${}$$
\vskip80pt

\centerline{\booktitlefont Complex Analytic and}\medskip
\centerline{\booktitlefont Differential Geometry}\bigskip\bigskip\bigskip

\centerline{\tenrmmfour Jean-Pierre Demailly}\bigskip\bigskip\bigskip\bigskip

\centerline{\tenrmmthree Universit\'e de Grenoble I}
\vskip8pt
\centerline{\tenrmmthree Institut Fourier, UMR 5582 du CNRS}
\vskip8pt
\centerline{\tenrmmthree 38402 Saint-Martin d'H\`eres, France}
\bigskip\bigskip\bigskip\bigskip
\centerline{Version of Thursday June 21, 2012}

\chapter{Table of Contents}{}
\setheadtitles{Table of Contents}
\opentocfile

\chapter{Chapter I}{Complex Differential Calculus and
Pseudoconvexity}
\begineightpoint
This introductive chapter is mainly a review of the basic tools and
concepts which will be employed in the rest of the book: differential
forms, currents, holomorphic and plurisubharmonic functions,
holomorphic convexity and pseudoconvexity. Our study of holomorphic
convexity is principally concentrated here on the case of domains in
$\bC^n$. The more powerful machinery needed for the study of general
complex varieties (sheaves, positive currents, hermitian differential
geometry) will be introduced in Chapters II to~V. Although our
exposition pretends to be almost self-contained, the reader is assumed
to have at least a vague familiarity with a few basic topics, such as
differential calculus, measure theory and distributions, holomorphic
functions of one complex variable, $\ldots\,$. Most of the necessary
background can be found in the books of \cite{Rudin 1966} and \cite{Warner
1971}; the basics of distribution theory can be found in Chapter~I of
\cite{H\"ormander 1963}. On the other hand, the reader who has already some
knowledge of complex analysis in several variables should probably
bypass this chapter.
\endeightpoint

\section{1}{Differential Calculus on Manifolds}
\subsection{1.A}{Differentiable Manifolds}
The notion of manifold is a natural extension of the notion of
submanifold defined by a set of equations in $\bR^n$. However,
as already observed by Riemann during the 19th century, it is
important to define the notion of a manifold in a flexible way,
without necessarily requiring that the underlying topological
space is embedded in an affine space. The precise formal definition
was first introduced by H.~Weyl in \cite{Weyl 1913}.

Let $m\in\bN$ and $k\in\bN\cup\{\infty,\omega\}$. We denote by
$\cC^k$ the class of functions which are $k$-times differentiable with
continuous derivatives if $k\ne\omega$, and by $C^\omega$ the class of
real analytic functions. A {\it differentiable manifold} $M$ of real
dimension $m$ and of class $\cC^k$ is a topological space (which
we shall always assume Hausdorff and separable, i.e.\ possessing a
countable basis of the topology), equipped with an atlas of class $\cC^k$ with
values in $\bR^m$. An {\it atlas} of class $\cC^k$
is a collection of homeomorphisms $\tau_\alpha:U_\alpha\longrightarrow V_\alpha$,
$\alpha\in I$, called {\it differentiable charts}, such that
$(U_\alpha)_{\alpha\in I}$ is an open covering of $M$ and $V_\alpha$ 
an open subset of $\bR^m$, and such that for all $\alpha,\beta\in I$
the {\it transition map}
$$\tau_{\alpha\beta}=\tau_\alpha\circ\tau_\beta^{-1}:
\tau_\beta(U_\alpha\cap U_\beta)\longrightarrow\tau_\alpha(U_\alpha\cap U_\beta)
\leqno(1.1)$$
is a $\cC^k$ diffeomorphism from an open subset of $V_\beta$ onto an
open subset of $V_\alpha$ (see Fig.~1). Then the components
$\tau_\alpha(x)=(x^\alpha_1,\ldots,x^\alpha_m)$ are called the {\it local
coordinates} on $U_\alpha$ defined by the chart $\tau_\alpha$~; they
are related by the transition relation $x^\alpha=\tau_{\alpha\beta}
(x^\beta)$.

\vbox{%
\input epsfiles/fig_1_1.tex
\vskip8mm
\centerline{{\bf Fig.~I-1} Charts and transition maps}}
\vskip5mm

If $\Omega\subset M$ is open and $s\in\bN\cup\{\infty,\omega\}$,
$0\le s\le k$, we denote by $C^s(\Omega,\bR)$ the set of
functions $f$ of class $C^s$ on $\Omega$, i.e.\ such that $f\circ
\tau_\alpha^{-1}$ is of class $C^s$ on $\tau_\alpha(U_\alpha\cap\Omega)$
for each $\alpha$~; if $\Omega$ is not open, $C^s(\Omega,\bR)$ is the
set of functions which have a $C^s$ extension to some neighborhood
of~$\Omega$.

A {\it tangent vector} $\xi$ at a point $a\in M$ is by definition
a differential operator acting on functions, of the type
$$C^1(\Omega,\bR)\ni f\longmapsto
\xi\cdot f=\sum_{1\le j\le m}\xi_j\,{\partial f\over\partial x_j}(a)$$
in any local coordinate system $(x_1,\ldots,x_m)$ on an open set
$\Omega\ni a$. We then simply write $\xi=\sum\xi_j\,\partial/\partial x_j$.
For every $a\in\Omega$, the $n$-tuple $(\partial/\partial x_j)_{1\le j\le m}$
is therefore a basis of the {\it tangent space} to $M$ at $a$, which we
denote by $T_{M,a}$. The {\it differential} of a function $f$ at $a$ is
the linear form on $T_{M,a}$ defined by
$$df_a(\xi)=\xi\cdot f=\sum\xi_j\,\partial f/\partial x_j(a),\qquad
\forall\xi\in T_{M,a}.$$
In particular $dx_j(\xi)=\xi_j$ and we may consequently write
$df=\sum(\partial f/\partial x_j)dx_j$.
From this, we see that
$(dx_1,\ldots,dx_m)$ is the dual basis of $(\partial/\partial x_1,\ldots,
\partial/\partial x_m)$ in the cotangent space $T^\star_{M,a}$.
The disjoint unions $T_M=\bigcup_{x\in M}T_{M,x}$ and $T^\star_M=
\bigcup_{x\in M}T^\star_{M,x}$ are called the {\it tangent} and 
{\it cotangent bundles} of~$M$.

If $\xi$ is a vector field of class $C^s$ over $\Omega$, that is, a map
$x\mapsto\xi(x)\in T_{M,x}$ such that $\xi(x)=\sum\xi_j(x)\,\partial/
\partial x_j$ has $C^s$ coefficients, and if $\eta$ is another
vector field of class $C^s$ with $s\ge 1$, the {\it Lie bracket} $[\xi,\eta]$
is the vector field such that
$$[\xi,\eta]\cdot f=\xi\cdot (\eta\cdot f)-\eta\cdot(\xi\cdot f).\leqno(1.2)$$
In coordinates, it is easy to check that
$$[\xi,\eta]=\sum_{1\le j,k\le m}\Big(\xi_j{\partial\eta_k\over
\partial x_j}-\eta_j{\partial\xi_k\over\partial x_j}\Big)\,
{\partial\over\partial x_k}.\leqno(1.3)$$

\subsection{1.B}{Differential Forms}
A differential form $u$ of degree $p$, or briefly a $p$-form
over $M$, is a map $u$ on $M$ with values $u(x)\in\Lambda^pT^\star_{M,x}$.
In a coordinate open set $\Omega\subset M$, a differential $p$-form
can be written
$$u(x)=\sum_{|I|=p}u_I(x)\,dx_I,$$
where $I=(i_1,\ldots,i_p)$ is a multi-index with integer components,
\hbox{$i_1<\ldots<i_p$} 
and $dx_I:=dx_{i_1}\wedge\ldots\wedge dx_{i_p}$. 
The notation $|I|$ stands for the number of components of $I$, and is
read {\it length} of $I$. For all integers $p=0,1,\ldots,m$ and
$s\in\bN\cup\{\infty\}$, $s\le k$, we denote by $C^s(M,\Lambda^pT^\star_M)$
the space of differential $p$-forms of class $C^s$, i.e.\ with $C^s$
coefficients $u_I$. Several natural operations on differential forms
can be defined.

\subsubsection{1.B.1}{Wedge Product.}
If $v(x)=\sum v_J(x)\,dx_J$ is a $q$-form, the {\it wedge product} of
$u$ and $v$ is the form of degree $(p+q)$ defined by
$$u\wedge v(x)=\sum_{|I|=p,|J|=q}u_I(x)v_J(x)\,dx_I\wedge dx_J.\leqno(1.4)$$

\subsubsection{1.B.2}{Contraction by a tangent vector.}
A $p$-form $u$ can be viewed as an antisymmetric $p$-linear form on~$T_M$.
If $\xi=\sum\xi_j\,\partial/\partial x_j$ is a tangent vector, we define
the {\it contraction} $\xi\ort u$ to be the differential form of degree
$p-1$ such that
$$(\xi\ort u)(\eta_1,\ldots,\eta_{p-1})=u(\xi,\eta_1,\ldots,\eta_{p-1})\leqno(1.5)$$
for all tangent vectors $\eta_j$. Then $(\xi,u)\longmapsto\xi\ort u$ is 
bilinear and we find easily
$${\partial\over\partial x_j}\ort dx_I=\cases{
0&if~~$j\notin I$,\cr
(-1)^{l-1}dx_{I\ssm\{j\}}&if~~$j=i_l\in I$.\cr}$$
A simple computation based on the above formula shows that
contraction by a tangent vector is a {\it derivation}, i.e.\
$$\xi\ort(u\wedge v)=(\xi\ort u)\wedge v+(-1)^{\deg u}u\wedge(\xi\ort v).
\leqno(1.6)$$

\subsubsection{1.B.3}{Exterior derivative.} This is the differential operator
$$d:C^s(M,\Lambda^pT^\star_M)\longrightarrow C^{s-1}(M,\Lambda^{p+1}T^\star_M)$$
defined in local coordinates by the formula
$$du=\sum_{|I|=p,\,1\le k\le m}{\partial u_I\over\partial x_k}\,
dx_k\wedge dx_I.\leqno(1.7)$$
Alternatively, one can define $du$ by its action on arbitrary vector fields
$\xi_0,\ldots,\xi_p$ on~$M$. The formula is as follows
$$\leqalignno{
du(\xi_0,\ldots,\xi_p)
&=\sum_{0\le j\le p}(-1)^j\xi_j\cdot u(\xi_0,\ldots,\wh{\xi_j},\ldots,\xi_p)\cr
&+\sum_{0\le j<k\le p}(-1)^{j+k}u([\xi_j,\xi_k],\xi_0,\ldots,\wh{\xi_j},\ldots,
\wh{\xi_k},\ldots,\xi_p).&(1.7')\cr}$$
The reader will easily check that (1.7) actually implies $(1.7')$. The
advantage of $(1.7')$ is that it does not depend on the choice of
coordinates, thus $du$ is intrinsically defined. The two basic
properties of the exterior derivative (again left to the reader) are:
$$\leqalignno{
&d(u\wedge v)=du\wedge v+(-1)^{\deg u}u\wedge dv,
\qquad\hbox{$(\,$\it Leibnitz' rule$\,)$}&(1.8)\cr
&d^2=0.&(1.9)\cr}$$
A form $u$ is said to be {\it closed} if $du=0$ and {\it exact} if $u$
can be written $u=dv$ for some form $v$.

\subsubsection{1.B.4}{De Rham Cohomology Groups.}
Recall that a cohomological complex $K^\bu=\bigoplus_{p\in\bZ}$ is 
a collection of modules $K^p$ over some ring, equipped with
differentials, i.e., linear maps $d^p:K^p\to K^{p+1}$ such that
$d^{p+1}\circ d^p=0$. The {\it cocycle, coboundary} and {\it cohomology
modules} $Z^p(K^\bu)$, $B^p(K^\bu)$ and $H^p(K^\bu)$ are defined
respectively by
$$\cases{Z^p(K^\bu)=\Ker d^p:K^p\to K^{p+1},&\quad $Z^p(K^\bu)\subset K^p$,\cr
B^p(K^\bu)=\Im d^{p-1}:K^{p-1}\to K^p,&\quad $B^p(K^\bu)\subset
Z^p(K^\bu)\subset K^p$,\cr
H^p(K^\bu)=Z^p(K^\bu)/B^p(K^\bu).&\cr}
\leqno(1.10)$$
Now, let $M$ be a differentiable manifold, say of class $\cC^\infty$ for simplicity.
The {\it De Rham complex} of $M$ is defined to be the complex
$K^p=\cC^\infty(M,\Lambda^pT^\star_M)$ of smooth differential forms,
together with the exterior derivative $d^p=d$ as differential,
and $K^p=\{0\}$, $d^p=0$ for $p<0$. 
We denote by $Z^p(M,\bR)$ the cocycles (closed $p$-forms) and
by $B^p(M,\bR)$ the coboundaries (exact $p$-forms). By convention
$B^0(M,\bR)=\{0\}$. The {\it De Rham cohomology group} of $M$ in
degree $p$ is 
$$H^p_\DR(M,\bR)=Z^p(M,\bR)/B^p(M,\bR).\leqno(1.11)$$
When no confusion with other types of cohomology groups may occur,
we sometimes denote these groups simply by $H^p(M,\bR)$. 
The symbol $\bR$ is used here to stress that we are considering real
valued $p$-forms; of course one can introduce a similar group
$H^p_\DR(M,\bC)$ for complex valued forms, i.e.\ forms with values in
$\bC\otimes\Lambda^p T^\star_M$. Then $H^p_\DR(M,\bC)=\bC\otimes
H^p_\DR(M,\bR)$ is the complexification of the real De Rham
cohomology group. It is clear that $H^0_\DR(M,\bR)$ can be identified
with the space of locally constant functions on~$M$, thus
$$H^0_\DR(M,\bR)=\bR^{\pi_0(X)},$$
where $\pi_0(X)$ denotes the set of connected components of~$M$.

Similarly, we introduce the De Rham cohomology groups with compact support
$$H^p_{\DR,c}(M,\bR)=Z^p_c(M,\bR)/B^p_c(M,\bR),\leqno(1.12)$$
associated with the De Rham complex $K^p=\cC^\infty_c(M,\Lambda^pT^\star_M)$
of smooth differential forms with compact support.

\subsubsection{1.B.5}{Pull-Back.}
If $F:M\longrightarrow M'$ is a differentiable map to another
manifold $M'$, $\dim_\bR M'=m'$, and if $v(y)=\sum v_J(y)\,dy_J$ is a
differential $p$-form on $M'$, the pull-back $F^\star v$ is the
differential $p$-form on $M$ obtained after making the substitution 
$y=F(x)$ in $v$, i.e.\
$$F^\star v(x)=\sum v_I\big(F(x)\big)\,dF_{i_1}\wedge\ldots\wedge dF_{i_p}.
\leqno(1.13)$$
If we have a second map $G:M'\longrightarrow M''$ and if $w$ is a differential form on
$M''$, then $F^\star(G^\star w)$ is obtained by means of the substitutions
$z=G(y)$, $y=F(x)$, thus
$$F^\star(G^\star w)=(G\circ F)^\star w.\leqno(1.14)$$
Moreover, we always have $d(F^\star v)=F^\star(dv)$. It follows that the
pull-back $F^\star$ is closed if $v$ is closed and exact if $v$ is exact.
Therefore $F^\star$ induces a morphism on the quotient spaces
$$F^\star:H^p_\DR(M',\bR)\longrightarrow H^p_\DR(M,\bR).\leqno(1.15)$$

\subsection{1.C}{Integration of Differential Forms}
A manifold $M$ is {\it orientable} if and only if there exists an atlas
$(\tau_\alpha)$ such that all transition maps $\tau_{\alpha\beta}$
preserve the orientation, i.e.\ have positive jacobian determinants.
Suppose that $M$ is oriented, that is, equipped with such an atlas. If
$u(x)=f(x_1,\ldots,x_m)\,dx_1\wedge\ldots\wedge dx_m$ is a continuous form of
ma\-ximum degree $m=\dim_\bR M$, with compact support in a coordinate
open set~$\Omega$, we set
$$\int_M u=\int_{\bR^m}f(x_1,\ldots,x_m)\,dx_1\ldots dx_m.\leqno(1.16)$$
By the change of variable formula, the result is independent of the
choice of coordinates, provided we consider only coordinates 
corresponding to the given orientation. When $u$ is an arbitrary
form with compact support, the definition of $\int_Mu$ is easily extended 
by means of a partition of unity with respect to coordinate open sets 
covering $\Supp u$. Let $F:M\longrightarrow M'$ be a diffeomorphism between oriented
manifolds and $v$ a volume form on $M'$. The change of variable formula 
yields
$$\int_M F^\star v=\pm\int_{M'}v\leqno(1.17)$$
according whether $F$ preserves orientation or not.

We now state Stokes' formula, which is basic in many contexts.
Let $K$ be a compact subset of $M$ with piecewise $C^1$ boun\-dary.
By this, we mean that for each point $a\in\partial K$ there are
coordinates $(x_1,\ldots,x_m)$ on a neighborhood $V$ of $a$, centered at $a$,
such that
$$K\cap V=\big\{x\in V\,;\, x_1\le 0,\ldots,x_l\le 0\big\}$$
for some index $l\ge 1$. Then $\partial K\cap V$ is a union of smooth
hypersurfaces with piecewise $C^1$ boundaries:
$$\partial K\cap V=\bigcup_{1\le j\le l}
\big\{x\in V\,;\,x_1\le 0,\ldots,x_j=0,\ldots,x_l\le 0\big\}.$$
At points of $\partial K$ where $x_j=0$, then $(x_1,\ldots,\wh{x_j},,\ldots,x_m)$
define coordinates on $\partial K$. We take the orientation of $\partial K$
given by these coordinates or the opposite one,
according to the sign $(-1)^{j-1}$. For any differential form $u$ of 
class $C^1$ and degree $m-1$ on $M$, we then have

\claim{(1.18) Stokes' formula}
$\qquad\displaystyle\int_{\partial K}u=\int_K du.$
\endclaim

The formula is easily checked by an explicit computation when $u$ has 
compact support in $V\,$: indeed if $u=\sum_{1\le j\le n}u_j\,dx_1\wedge
\ldots\wh{dx_j}\ldots dx_m$ and $\partial_jK\cap V$ is the part of $\partial
K\cap V$ where $x_j=0$, a partial integration with respect to $x_j$ yields
$$\eqalign{
&\int_{\partial_jK\cap V}u_j\,dx_1\wedge\ldots\wh{dx_j}\ldots dx_m
=\int_V{\partial u_j\over\partial x_j}\,dx_1\wedge\ldots dx_m,\cr
&\int_{\partial K\cap V} u=\sum_{1\le j\le m}(-1)^{j-1}
\int_{\partial_jK\cap V}u_j\,dx_1\wedge\ldots\wh{dx_j}\ldots\wedge dx_m
=\int_V du.\cr}$$
The general case follows by a partition of unity. In particular, if $u$ has
compact support in $M$, we find $\int_M du=0$ by choosing $K\supset\Supp u$.

\subsection{1.D}{Homotopy Formula and Poincar\'e Lemma}
Let $u$ be a differential form on $[0,1]\times M$. For $(t,x)\in[0,1]
\times M$, we write
$$u(t,x)=\sum_{|I|=p}u_I(t,x)\,dx_I+\sum_{|J|=p-1}\wt u_J(t,x)\,dt\wedge dx_J.$$
We define an operator
$$\leqalignno{
&K:C^s([0,1]\times M,\Lambda^pT^\star_{[0,1]\times M})\longrightarrow
C^s(M,\Lambda^{p-1}T^\star_M)\cr
&Ku(x)=\sum_{|J|=p-1}\Big(\int_0^1\wt u_J(t,x)\,dt\Big)dx_J&(1.19)\cr}$$
and say that $Ku$ is the form obtained by integrating $u$ along $[0,1]$. 
A computation of the operator $dK+Kd$ shows that all terms involving
partial derivatives $\partial \wt u_J/\partial x_k$ cancel, hence
$$\leqalignno{
Kdu+dKu&=\sum_{|I|=p}\Big(\int_0^1{\partial u_I\over\partial t}(t,x)
\,dt\Big)dx_I=\sum_{|I|=p}\big(u_I(1,x)-u_I(0,x)\big)dx_I,\cr
\qquad Kdu+dKu&=i_1^\star u-i_0^\star u,&(1.20)\cr}$$
where $i_t:M\to[0,1]\times M$ is the injection $x\mapsto(t,x)$.

\claim{(1.20) Corollary} Let $F,G:M\longrightarrow M'$ be $\cC^\infty$ maps. Suppose that
$F,G$ are smoothly homotopic, i.e.\ that there exists a $\cC^\infty$ map
$H:[0,1]\times M\longrightarrow M'$ such that $H(0,x)=F(x)$ and $H(1,x)=G(x)$. Then
$$F^\star=G^\star:H^p_\DR(M',\bR)\longrightarrow H^p_\DR(M,\bR).$$
\endclaim

\proof{Proof} If $v$ is a $p$-form on $M'$, then
$$\eqalign{
G^\star v-F^\star v&=(H\circ i_1)^\star v-(H\circ i_0)^\star v=
i_1^\star(H^\star v)-i_0^\star(H^\star v)\cr
&=d(KH^\star v)+KH^\star(dv)\cr}$$
by (1.20) applied to $u=H^\star v$. If $v$ is closed, then $F^\star v$
and $G^\star v$ differ by an exact form, so they define the same class in
$H^p_\DR(M,\bR)$.\qed
\endproof

\claim{(1.21) Corollary} If the manifold $M$ is {\it contractible},
i.e.\ if there is a smooth
homotopy $H:[0,1]\times M\to M$ from a constant map $F:M\to\{x_0\}$ to
$G=\Id_X$, then $H^0_\DR(M,\bR)=\bR$ and $H^p_\DR(M,\bR)=0$ for
$p\ge 1$.
\endclaim

\proof{Proof} $F^\star$ is clearly zero in degree $p\ge 1$, while
$F^\star:H^0_\DR(M,\bR)\buildo\simeq\over\longrightarrow\bR$ is induced by the
evaluation map $u\mapsto u(x_0)$. The conclusion then follows from the
equality $F^\star=G^\star=\Id$ on cohomology groups.\qed
\endproof

\claim{(1.22) Poincar\'e lemma} Let $\Omega\subset\bR^m$ be a
starshaped open set. If a form $v=\sum v_Idx_I\in
C^s(\Omega,\Lambda^pT^\star_\Omega)$, $p\ge 1$, satisfies $dv=0$, there
exists a form $u\in C^s(\Omega,\Lambda^{p-1}T^\star_\Omega)$ such that
$du=v$.
\endclaim

\proof{Proof} Let $H(t,x)=tx$ be the homotopy between the identity map 
$\Omega\to\Omega$ and the constant map $\Omega\to\{0\}$. By the above formula
$$d(KH^\star v)=G^\star v-F^\star v=\cases{
v-v(0)&if~~$p=0$,\cr
v&if~~$p\ge 1$.\cr}$$
Hence $u=KH^\star v$ is the $(p-1)$-form we are looking for. An explicit
computation based on (1.19) easily gives
$$u(x)={}\!\!\sum_{\scriptstyle |I|=p\atop\scriptstyle 1\le k\le p}\!\!
\Big(\int_0^1t^{p-1}v_I(tx)\,dt\Big)
(-1)^{k-1}x_{i_k}dx_{i_1}\wedge\ldots\wh{dx_{i_k}}\ldots
\wedge dx_{i_p}.\leqno(1.23)$$
\endproof

\section{2}{Currents on Differentiable Manifolds}
\subsection{2.A}{Definition and Examples}
Let $M$ be a $\cC^\infty$ differentiable manifold, $m=\dim_\bR M$. All the
manifolds considered in Sect.~2 will be assumed to be oriented. 
We first introduce a topology on the space of differential forms $C^s(M,
\Lambda^pT^\star_M)$. Let $\Omega\subset M$ be a coordinate open set and
$u$ a $p$-form on $M$, written $u(x)=\sum u_I(x)\,dx_I$ on $\Omega$.
To every compact subset $L\subset\Omega$ and every integer $s\in\bN$, we
associate a seminorm
$$p^s_L(u)=\sup_{x\in L}\,\max_{|I|=p,|\alpha|\le s}|D^\alpha u_I(x)|,
\leqno(2.1)$$
where $\alpha=(\alpha_1,\ldots,\alpha_m)$ runs over $\bN^m$ and $D^\alpha=
\partial^{|\alpha|}/\partial x_1^{\alpha_1}\ldots\partial x_m^{\alpha_m}$
is a derivation of order $|\alpha|=\alpha_1+\cdots+\alpha_m$. This type of
multi-index, which will always be denoted by Greek letters, should not
be confused with multi-indices of the type $I=(i_1,\ldots,i_p)$ introduced 
in Sect.~1. 

\claim{(2.2) Definition} We introduce as follows spaces of $p$-forms
on manifolds.
\smallskip
\item{\rm a)} We denote by $\cE^p(M)$ $\big($resp.\ 
${}^s\cE^p(M)\big)$ the space $\cC^\infty(M,\Lambda^pT^\star_M)$ $\big($resp.\
the space $C^s(M,\Lambda^pT^\star_M)\big)$, equipped with the topology
defined by all seminorms $p^s_L$ when $s$, $L$, $\Omega$ vary $($resp.\
when $L$, $\Omega$ vary$)$.
\smallskip
\item{\rm b)} If $K\subset M$ is a compact subset, $\cD^p(K)$ will denote 
the subspace of elements $u\in\cE^p(M)$ with support contained in $K$, 
together with the induced topo\-logy; $\cD^p(M)$ will stand for the set of
all elements with compact support, i.e.\ $\cD^p(M):=\bigcup_K\cD^p(K)$.
\smallskip
\item{\rm c)}The spaces of $C^s$-forms ${}^s\cD^p(K)$ and ${}^s\cD^p(M)$ 
are defined similarly.
\vskip0pt
\endclaim

Since our manifolds are assumed to be separable, the topology
of $\cE^p(M)$ can be defined by means of a countable set of seminorms
$p^s_L$, hence $\cE^p(M)$ (and likewise ${}^s\cE^p(M)$) is a Fr\'echet
space. The topology of ${}^s\cD^p(K)$ is induced by any finite
set of seminorms $p^s_{K_j}$ such that the compact sets $K_j$ cover $K$~;
hence ${}^s\cD^p(K)$ is a Banach space.
It should be observed however that $\cD^p(M)$ is not a Fr\'echet space;
in fact $\cD^p(M)$ is dense in $\cE^p(M)$ and thus non complete for the
induced topology. According to \cite{De~Rham 1955} spaces of {\it currents}
are defined as the topological duals of the above spaces, in analogy with
the usual definition of distributions. 

\claim{(2.3) Definition} The space of currents of dimension $p$ 
$($or degree $m-p)$ on $M$ is the space $\cD'_p(M)$ of linear forms
$T$ on $\cD^p(M)$ such that the
restriction of $T$ to all subspaces $\cD^p(K)$, $K\compact M$,
is continuous. The degree is indicated by raising the index, hence we set
$$\cD^{\prime\,m-p}(M)=\cD'_p(M):=\hbox{topological dual~}
\big(\cD^p(M)\big)'.$$
The space ${}^s\cD'_p(M)={}^s\cD^{\prime\,m-p}(M):=\big({}^s\cD^p(M)\big)'$
is defined similarly and is called the space of currents of order $s$ on $M$.
\endclaim

In the sequel, we let $\langle T,u\rangle$ be the pairing between a
current $T$ and a {\it test form} $u\in\cD^p(M)$.
It is clear that ${}^s\cD'_p(M)$ can be identified with the subspace 
of currents $T\in\cD'_p(M)$ which are continuous for the seminorm $p^s_K$ 
on $\cD^p(K)$ for every compact set $K$ contained in a coordinate patch
$\Omega$. The {\it support} of $T$, denoted
Supp$\,T$, is the smallest closed subset $A\subset M$ such that the restriction
of $T$ to $\cD^p(M\ssm A)$ is zero. The topological dual $\cE'_p(M)$
can be identified with 
the set of currents of $\cD'_p(M)$ with compact support: indeed,
let $T$ be a linear form on $\cE^p(M)$ such that 
$$|\langle T,u\rangle|\le C\max\{p^s_{K_j}(u)\}$$
for some $s\in\bN$, $C\ge 0$ and a finite number of compact sets $K_j$~;
it follows that Supp$\,T\subset\bigcup K_j$. Conversely let $T\in\cD'_p(M)$
with support in a compact set $K$. Let $K_j$ be compact patches such
that $K$ is contained in the interior of $\bigcup K_j$ and
$\psi\in\cD(M)$ equal to $1$ on $K$ with Supp$\,\psi\subset\bigcup K_j$.
For $u\in\cE^p(M)$, we define $\langle T,u\rangle=\langle T,\psi u\rangle$~;
this is independent of $\psi$ and the resulting $T$ is clearly
continuous on $\cE^p(M)$.  
The terminology used for the dimension and degree of a current
is justified by the following two examples. 

\claim{(2.4) Example} \rm Let $Z\subset M$ be a closed oriented 
submanifold of $M$ of dimension $p$ and class $C^1$~; $Z$ may have a boundary 
$\partial Z$. The {\it current of integration} over $Z$, denoted $[Z]$,
is defined by
$$\langle[Z],u\rangle=\int_Z u,~~~~u\in{}^0\cD^p(M).$$
It is clear that $[Z]$ is a current of order $0$ on $M$ and that~
$\Supp [Z]=Z$. Its dimension is $p=\dim Z$.
\endclaim

\claim{(2.5) Example} \rm If $f$ is a differential form of degree $q$
on $M$ with $L^1_\loc$ coefficients, we can associate to $f$ the
current of dimension $m-q$~:
$$\langle T_f,u\rangle=\int_M f\wedge u,~~~~u\in{}^0\cD^{m-q}(M).$$
$T_f$ is of degree $q$ and of order $0$. The correspondence 
$f\longmapsto T_f$ is injective.
In the same way $L^1_\loc$ functions on $\bR^m$ are identified to
distributions, we will identify $f$ with its image
$T_f\in{}^0\cD^{\prime\,q}(M)={}^0\cD'_{m-q}(M)$.
\endclaim

\subsection{2.B}{Exterior Derivative and Wedge Product}
\subsubsection{2.B.1}{Exterior Derivative.}
Many of the operations available for differential forms can be extended
to currents by simple duality arguments. Let $T\in{}^s\cD^{\prime\,q}(M)=
{}^s\cD'_{m-p}(M)$. The {\it exterior derivative}
$$dT\in{}^{s+1}\cD^{\prime\,q+1}(M)={}^{s+1}\cD'_{m-q-1}$$
is defined by
$$\langle dT,u\rangle=(-1)^{q+1}\,\langle T,du\rangle,~~~~
u\in{}^{s+1}\cD^{m-q-1}(M).\leqno(2.6)$$
The continuity of the linear form $dT$ on ${}^{s+1}\cD^{m-q-1}(M)$ follows
from the continuity of the map \hbox{$d:{}^{s+1}\cD^{m-q-1}(K)\longrightarrow
{}^s\cD^{m-q}(K)$}. For all forms $f\in {}^1\cE^q(M)$ and $u\in\cD^{m-q-1}(M)$, Stokes' formula 
implies
$$0=\int_M d(f\wedge u)=\int_M df\wedge u+(-1)^q\,f\wedge du,$$
thus in example (2.5) one actually has $dT_f=T_{df}$ as it should be.
In example (2.4), another application of Stokes' formula yields
$\int_Z du=\int_{\partial Z}u$, therefore $\langle[Z],du\rangle=
\langle[\partial Z],u\rangle$ and
$$d[Z]=(-1)^{m-p+1}[\partial Z].\leqno(2.7)$$

\subsubsection{2.B.2}{Wedge Product.}
For $T\in{}^s\cD^{\prime\,q}(M)$ and $g\in{}^s\cE^r(M)$, the wedge product
$T\wedge g\in{}^s\cD^{\prime\,q+r}(M)$ is defined by
$$\langle T\wedge g,u\rangle=\langle T,g\wedge u\rangle,~~~~
u\in{}^s\cD^{m-q-r}(M).\leqno(2.8)$$
This definition is licit because $u\mapsto g\wedge u$ is continuous in the 
$C^s$-topology. The relation 
$$d(T\wedge g)=dT\wedge g+(-1)^{{\rm deg}\,T}T\wedge dg$$
is easily verified from the definitions.
 
\claim{(2.9) Proposition} Let $(x_1,\ldots,x_m)$ be a coordinate system on
an open subset $\Omega\subset M$. Every current $T\in{}^s\cD^{\prime\,q}(M)$
of degree $q$ can be written in a unique way
$$T=\sum_{|I|=q}T_I\,dx_I~~~~\hbox{\rm on}~~\Omega,$$
where $T_I$ are distributions of order $s$ on $\Omega$, considered as
currents of degree~$0$.
\endclaim

\proof{Proof} If the result is true, for all $f\in{}^s\cD^0(\Omega)$ we must
have
$$\langle T,f\,dx_{\complement I}\rangle
=\langle T_I,dx_I\wedge f\,dx_{\complement I}\rangle
=\varepsilon(I,\complement I)\,\langle T_I,f\,dx_1\wedge\ldots\wedge
dx_m\rangle,$$
where $\varepsilon(I,\complement I)$ is the signature of the permutation
$(1,\ldots,m)\longmapsto(I,\complement I)$. Conversely, this can be taken as
a definition of the coefficient $T_I\,$:
$$T_I(f)=\langle T_I,f\,dx_1\wedge\ldots\wedge dx_m\rangle:=
\varepsilon(I,\complement I)\,\langle T,f\,dx_{\complement I}
\rangle,~~f\in{}^s\cD^0(\Omega).\leqno(2.10)$$
Then $T_I$ is a distribution of order $s$ and it is easy to check that
$T=\sum T_I\,dx_I$.\qed
\endproof

In particular, currents of order $0$ on $M$ can be considered as differential 
forms with measure coefficients. In order to unify the notations concerning
forms and currents, we set
$$\langle T,u\rangle=\int_M T\wedge u$$
whenever $T\in{}^s\cD'_p(M)={}^s\cD^{\prime\,m-p}(M)$ and $u\in{}^s\cE^p(M)$
are such that Supp$\,T\cap{\rm Supp}\,u$ is compact. This convention is
made so that the notation becomes compatible with the identification of a
form $f$ to the current $T_f$.

\subsection{2.C}{Direct and Inverse Images}
\subsubsection{2.C.1}{Direct Images.} 
Assume now that $M_1$, $M_2$ are oriented differentiable manifolds of
respective dimensions $m_1$, $m_2$, and that
$$F:M_1\longrightarrow M_2\leqno(2.11)$$
is a $\cC^\infty$ map. The pull-back morphism
$${}^s\cD^p(M_2)\longrightarrow{}^s\cE^p(M_1),\qquad u\longmapsto F^\star u
\leqno(2.12)$$
is continuous in the $C^s$ topology and we have Supp$\,F^\star u
\subset F^{-1}({\rm Supp}\,u)$, but in general Supp$\,F^\star u$
is not compact. If $T\in{}^s\cD'_p(M_1)$ is such that the restriction
of $F$ to Supp$\,T$ is {\it proper},
i.e.\ if ${\rm Supp}\,T\cap F^{-1}(K)$ is compact for every compact subset
$K\subset M_2$, then the linear form $u\longmapsto\langle T,F^\star 
u\rangle$ is well defined and continuous on ${}^s\cD^p(M_2)$. There exists 
therefore a unique current denoted $F_\star T\in{}^s\cD'_p(M_2)$,
called {\it the direct image} of $T$ by $F$, such that
$$\langle F_\star T,u\rangle=\langle T,F^\star u\rangle,~~~~
\forall u\in{}^s\cD^p(M_2).\leqno(2.13)$$
We leave the straightforward proof of the following properties to the reader.

\claim{(2.14) Theorem} For every $T\in{}^s\cD'_p(M_1)$ such that
$F_{\restriction {\rm Supp}\,T}$ is proper, the direct image
$F_\star T\in{}^s\cD'_p(M_2)$ is such that 
\medskip
\item{\rm a)} {\rm Supp}$\,F_\star T\subset F({\rm Supp}\,T)~;$
\smallskip
\item{\rm b)} $d(F_\star T)=F_\star(dT)~;$
\smallskip
\item{\rm c)} $\,F_\star(T\wedge F^\star g)=(F_\star T)
\wedge g$,~~~~$\forall g\in {}^s\cE^q(M_2,\bR)~;$
\smallskip
\item{\rm d)} If $G:M_2\longrightarrow M_3$ is a $\cC^\infty$ map such that
$(G\circ F)_{\restriction{\rm Supp}\,T}$ is proper, then
$$G_\star(F_\star T)=(G\circ F)_\star T.$$
\endclaim

\claim{(2.15) Special case} {\rm Assume that $F$ is a submersion, i.e.\ 
that $F$ is surjective and that for every $x\in M_1$ the differential 
map $d_xF:T_{M_1,x}\longrightarrow T_{M_2,F(x)}$ is surjective. Let $g$ be a differential
form of degree $q$ on $M_1$, with $L^1_\loc$ coefficients, such that 
$F_{\restriction{\rm Supp}\,g}$ is proper. We claim that
$F_\star g\in{}^0\cD'_{m_1-q}(M_2)$ 
is the form of degree $q-(m_1-m_2)$ obtained from $g$ by 
integration along the fibers of $F$, also denoted
$$F_\star g(y)=\int_{z\in F^{-1}(y)}g(z).$$
\vskip2mm

\vbox{%
\input epsfiles/fig_1_2.tex
\vskip6mm
\centerline{{\bf Fig.~I-2} Local description of a submersion as a projection.}}
\vskip5mm

\noindent
In fact, this assertion is 
equivalent to the following generalized form of Fubini's theorem:
$$\int_{M_1}g\wedge F^\star u=\int_{y\in M_2}\Big(\int_{z\in F^{-1}(y)}
g(z)\Big)\wedge u(y),~~~~\forall u\in{}^0\cD^{m_1-q}(M_2).$$
By using a partition of unity on $M_1$ and the constant rank theorem,
the verification of this formula is easily reduced to the case where
$M_1=A\times M_2$ and $F={\rm pr}_2$, cf.\ Fig.~2. The fibers $F^{-1}(y)
\simeq A$ have to be oriented in such a way that the orientation of $M_1$
is the product of the orientation of $A$ and $M_2$. Let us write
$r=\dim A=m_1-m_2$ and let $z=(x,y)\in A\times M_2$ be any point of $M_1$.
The above formula becomes
$$\int_{A\times M_2}g(x,y)\wedge u(y)=\int_{y\in M_2}\Big(
\int_{x\in A}g(x,y)\Big)\wedge u(y),$$
where the direct image of $g$ is computed from $g=\sum g_{I,J}(x,y)\,dx_I
\wedge dy_J$, $|I|+|J|=q$, by the formula
$$\leqalignno{
\qquad F_\star g(y)&=\int_{x\in A}g(x,y)&(2.16)\cr
&=\sum_{|J|=q-r}\Big(\int_{x\in A}g_{(1,\ldots,r),J}(x,y)\,
dx_1\wedge\ldots\wedge dx_r\Big)dy_J.\cr}$$
In this situation, we see that $F_\star g$ has $L^1_\loc$ coefficients
on $M_2$ if $g$ is $L^1_\loc$ on $M_1$, and that the map
$g\longmapsto F_\star g$ is continuous in the $C^s$ topology.}
\endclaim

\claim{(2.17) Remark} \rm If $F:M_1\longrightarrow M_2$ is a diffeomorphism, then we 
have
$F_\star g=\pm(F^{-1})^\star g$ according whether $F$ preserves 
the orientation or not. In fact formula (1.17) gives
$$\langle F_\star g,u\rangle=\int_{M_1}g\wedge F^\star u=\pm\int_{M_2}
(F^{-1})^\star
(g\wedge F^\star u)=\pm\int_{M_2}(F^{-1})^\star g\wedge u.$$
\endclaim

\subsubsection{2.C.2}{Inverse Images.} 
Assume that $F:M_1\longrightarrow M_2$ is a submersion. As a consequence of the
continuity statement after (2.16), one can always define the inverse image 
$F^\star T\in{}^s\cD^{\prime\,q}(M_1)$ of a current $T\in{}^s\cD^{\prime\,
q}(M_2)$ by
$$\langle F^\star T,u\rangle=\langle T,F_\star u\rangle,~~~~
u\in{}^s\cD^{q+m_1-m_2}(M_1).$$
Then $\dim F^\star T=\dim T+m_1-m_2$ and Th.~2.14 yields
the formulas:
$$d(F^\star T)=F^\star(dT),~~~~F^\star(T\wedge g)=
F^\star T\wedge F^\star g,~~~~\forall g\in{}^s\cD^\bu(M_2).\leqno(2.18)$$
Take in particular $T=[Z]$, where $Z$ is an oriented $C^1$-submanifold of 
$M_2$. Then $F^{-1}(Z)$ is a submanifold of $M_1$ and has a natural 
orientation given by the isomorphism
$$T_{M_1,x}/T_{F^{-1}(Z),x}\longrightarrow T_{M_2,F(x)}/T_{Z,F(x)},$$
induced by $d_xF$ at every point $x\in Z$. We claim that
$$F^\star[Z]=[F^{-1}(Z)].\leqno(2.19)$$
Indeed, we have to check that $\int_ZF_\star u=\int_{F^{-1}(Z)}u$
for every $u\in{}^s\cD^\bullet(M_1)$. By using a partition of unity on $M_1$,
we may again assume $M_1=A\times M_2$ and $F={\rm pr}_2$. The above equality
can be written
$$\int_{y\in Z}F_\star u(y)=\int_{(x,y)\in A\times Z}u(x,y).$$
This follows precisely from (2.16) and Fubini's theorem. 

\subsubsection{2.C.3}{Weak Topology.}
The weak topology on $\cD'_p(M)$ is the topology defined by the collection
of seminorms $T\longmapsto|\langle T,f\rangle|$ for all $f\in\cD^p(M)$.
With respect to the weak topology, all the operations 
$$T\longmapsto dT,~~~~T\longmapsto T\wedge g,~~~~
T\longmapsto F_\star T,~~~~T\longmapsto F^\star T\leqno(2.20)$$ 
defined above are continuous. A set $B\subset\cD'_p(M)$ is bounded for
the weak topology (weakly bounded for short) if and only if $\langle
T,f\rangle$ is bounded when $T$ runs over $B$, for every fixed 
$f\in\cD^p(M)$. The standard Banach-Alaoglu theorem implies that 
every weakly bounded closed subset $B\subset\cD'_p(M)$ is weakly compact.

\subsection{2.D}{Tensor Products, Homotopies and Poincar\'e Lemma}
\subsubsection{2.D.1}{Tensor Products.}
If $S$, $T$ are currents on manifolds $M$, $M'$ there exists a unique current 
on $M\times M'$, denoted $S\otimes T$ and defined in a way analogous to the
tensor product of distributions, such that for all
$u\in\cD^\bu(M)$ and $v\in\cD^\bu(M')$
$$\langle S\otimes T,{\rm pr}_1^\star u\wedge{\rm pr}_2^\star v\rangle=
(-1)^{{\rm deg}\,T\,{\rm deg}\,u}\langle S,u\rangle\,\langle T,v\rangle.
\leqno(2.21)$$
One verifies easily that~ $d(S\otimes T)=dS\otimes T+(-1)^{{\rm deg}\,S}
S\otimes dT$.

\subsubsection{2.D.2}{Homotopy Formula.}  
Assume that $H:[0,1]\times M_1\longrightarrow M_2$ is a $\cC^\infty$ homotopy from
$F(x)=H(0,x)$ to $G(x)=H(1,x)$ and that $T\in\cD'_\bu(M_1)$ is a
current such that $H_{\restriction[0,1]\times{\rm Supp}\,T}$ is proper. If
$[0,1]$ is considered as the current of degree $0$ on $\bR$ associated to its
characteristic function, we find $d[0,1]=\delta_0-\delta_1$, thus
$$\eqalign{
d\big(H_\star([0,1]\otimes T)\big)
&=H_\star(\delta_0\otimes T-\delta_1\otimes T+[0,1]\otimes dT)\cr
&=F_\star T-G_\star T+H_\star([0,1]\otimes dT).\cr}$$
Therefore we obtain the {\it homotopy formula}
$$F_\star T-G_\star T=d\big(H_\star([0,1]\otimes T)\big)-
H_\star([0,1]\otimes dT).\leqno(2.22)$$
When $T$ is closed, i.e.\ $dT=0$, we see that $F_\star T$ and 
$G_\star T$ are cohomologous on $M_2$, i.e.\ they differ by an
exact current $dS$.

\subsubsection{2.D.3}{Regularization of Currents.} 
Let $\rho\in \cC^\infty(\bR^m)$ be a function with support in $B(0,1)$,
such that $\rho(x)$ depends only on $|x|=(\sum|x_i|^2)^{1/2}$, 
$\rho\ge 0$ and $\int_{\bR^m}\rho(x)\,dx=1$.  We associate
to $\rho$ the family of functions $(\rho_\varepsilon)$ such that
$$\rho_\varepsilon(x)={1\over\varepsilon^m}\,\rho\Big({x\over\varepsilon}\Big),
~~~~\hbox{\rm Supp}\,\rho_\varepsilon\subset B(0,\varepsilon),~~~~
\int_{\bR^m}\rho_\varepsilon(x)\,dx=1.\leqno(2.23)$$
We shall refer to this construction by saying that $(\rho_\varepsilon)$ is a
{\it family of smoothing kernels}. For every current $T=\sum T_I\,dx_I$ on an
open subset $\Omega\subset\bR^m$, the family of smooth forms 
$$T\star\rho_\varepsilon=\sum_I~(T_I\star\rho_\varepsilon)\,dx_I,$$
defined on $\Omega_\varepsilon=\{x\in\bR^m~;~d(x,\complement\Omega)>
\varepsilon\}$, converges weakly to $T$ as $\varepsilon$ tends to $0$.
Indeed, $\langle T\star\rho_\varepsilon,f\rangle=\langle T,
\rho_\varepsilon\star f\rangle$ and $\rho_\varepsilon\star f$ converges to
$f$ in $\cD^p(\Omega)$ with respect to all seminorms~$p^s_K$.

\subsubsection{2.D.4}{Poincar\'e Lemma for Currents.}
Let $T\in{}^s\cD^{\prime\,q}(\Omega)$
be a closed current on an open set $\Omega\subset\bR^m$. We first show 
that $T$ is cohomologous to a smooth form. In fact, let $\psi\in \cC^\infty(\bR^m)$
be a cut-off function such that $\Supp\psi\subset\ol\Omega$, $0<\psi\le 1$
and $|d\psi|\le 1$ on $\Omega$. For any vector $v\in B(0,1)$ we set
$$F_v(x)=x+\psi(x)v.$$
Since $x\mapsto\psi(x)v$ is a contraction, $F_v$ is a diffeomorphism
of $\bR^m$ which leaves $\complement\Omega$ invariant pointwise, so
$F_v(\Omega)=\Omega$. This diffeomorphism is homotopic to the
identity through the homotopy $H_v(t,x)=F_{tv}(x):[0,1]\times\Omega
\longrightarrow\Omega$ which is proper for every $v$. Formula (2.22) implies
$$(F_v)_\star T-T=d\big((H_v)_\star([0,1]\otimes T)\big).$$
After averaging with a smoothing kernel $\rho_\varepsilon(v)$ we get
$\Theta-T=dS$ where
$$\Theta=\int_{B(0,\varepsilon)}(F_v)_\star T\,\rho_\varepsilon(v)\,dv,
~~~~S=\int_{B(0,\varepsilon)}(H_v)_\star([0,1]\otimes T)
\,\rho_\varepsilon(v)\,dv.$$
Then $S$ is a current of the same order $s$ as $T$ and $\Theta$ is smooth.
Indeed, for $u\in\cD^p(\Omega)$ we have
$$\langle\Theta,u\rangle=\langle T,u_\varepsilon\rangle~~~~\hbox{\rm where}~~
u_\varepsilon(x)=\int_{B(0,\varepsilon)}F_v^\star u(x)\,
\rho_\varepsilon(v)\,dv~;$$
we can make a change of variable $z=F_v(x)\Leftrightarrow
v=\psi(x)^{-1}(z-x)$ in the last integral and perform derivatives on
$\rho_\varepsilon$ to see that each seminorm $p^t_K(u_\varepsilon)$ is
controlled by the sup norm of $u$.  Thus $\Theta$ and all its
derivatives are currents of order $0$, so $\Theta$ is smooth.  Now we
have $d\Theta=0$ and by the usual Poincar\'e lemma (1.22) applied to
$\Theta$ we obtain

\claim{(2.24) Theorem} Let $\Omega\subset\bR^m$ be a starshaped open subset
and $T\in{}^s\cD^{\prime\,q}(\Omega)$ a current of degree $q\ge 1$ and order
$s$ such that $dT=0$. There exists a current $S\in{}^s\cD^{\prime\,q-1}
(\Omega)$ of degree $q-1$ and order $\le s$ such that $dS=T$ on $\Omega$.\qed
\endclaim

\section{3}{Holomorphic Functions and Complex Manifolds}
\subsection{3.A}{Cauchy Formula in One Variable}
We start by recalling a few elementary facts in one complex variable
theory. Let $\Omega\subset\bC$ be an open set and let $z=x+\ii y$
be the complex variable, where $x,y\in\bR$. If $f$ is a function of
class $C^1$ on $\Omega$, we have
$$df={\partial f\over\partial x}\,dx+{\partial f\over\partial y}\,dy
={\partial f\over\partial z}\,dz+{\partial f\over\partial\ol z}\,d\ol z$$
with the usual notations
$${\partial\over\partial z}={1\over 2}\Big({\partial\over\partial x}
-\ii{\partial\over\partial y}\Big),~~~~
{\partial\over\partial\ol z}={1\over 2}\Big({\partial\over\partial x}
+\ii{\partial\over\partial y}\Big).\leqno(3.1)$$
The function $f$ is holomorphic on $\Omega$ if $df$ is $\bC$-linear,
that is, $\partial f/\partial\ol z=0$.

\claim{(3.2) Cauchy formula} Let $K\subset\bC$ be a compact set
with piecewise $C^1$ boun\-dary $\partial K$. Then for every 
$f\in C^1(K,\bC)$
$$f(w)={1\over 2\pi\ii}\int_{\partial K}{f(z)\over z-w}\,dz-
\int_K{1\over\pi(z-w)}\,{\partial f\over\partial\ol z}\,d\lambda(z),~~~~
w\in K^\circ$$
where $d\lambda(z)={\ii\over 2}dz\wedge d\ol z=dx\wedge dy$ is the Lebesgue 
measure on $\bC$.
\endclaim

\proof{Proof} Assume for simplicity $w=0$. As the function $z\mapsto 1/z$ is locally
integrable at $z=0$, we get
$$\eqalign{
\int_K{1\over\pi z}\,{\partial f\over\partial\ol z}\,d\lambda(z)
&=\lim_{\varepsilon\to 0}\int_{K\ssm D(0,\varepsilon)}
 {1\over\pi z}\,{\partial f\over\partial\ol z}\,{\ii\over 2}dz\wedge d\ol z\cr
&=\lim_{\varepsilon\to 0}\int_{K\ssm D(0,\varepsilon)}
  d\Big[{1\over 2\pi\ii}\,f(z)\,{dz\over z}\Big]\cr
&={1\over 2\pi\ii}\int_{\partial K}f(z)\,{dz\over z}-
  \lim_{\varepsilon\to 0}{1\over 2\pi\ii}\int_{\partial D(0,\varepsilon)}
  f(z)\,{dz\over z}\cr}$$
by Stokes' formula. The last integral is equal to
${1\over 2\pi}\int_0^{2\pi}f(\varepsilon e^{\ii\theta})\,d\theta$ and 
converges to $f(0)$ as $\varepsilon$ tends to $0$.\qed
\endproof

When $f$ is holomorphic on $\Omega$, we get the usual Cauchy formula
$$f(w)={1\over 2\pi\ii}\int_{\partial K}{f(z)\over z-w}\,dz,~~~~
w\in K^\circ,\leqno(3.3)$$
from which many basic properties of holomorphic functions can be derived:
power and Laurent series expansions, Cauchy residue formula, $\ldots$
Another interesting consequence is:

\claim{(3.4) Corollary} The $L^1_\loc$ function $E(z)=1/\pi z$ is
a fundamental solution of the operator $\partial/\partial\ol z$ on $\bC$,
i.e.\ $\partial E/\partial\ol z=\delta_0$ $($Dirac measure at $0)$. As a
consequence, if $v$ is a distribution with compact support in $\bC$, then
the convolution $u=(1/\pi z)\star v$ is a solution of the equation
$\partial u/\partial\ol z=v$.
\endclaim

\proof{Proof} Apply (3.2) with $w=0$, $f\in\cD(\bC)$ and $K\supset{\rm
Supp}\,f$, so that $f=0$ on the boundary $\partial K$ and $f(0)=\langle
1/\pi z,-\partial f/\partial\ol z\rangle$.\qed
\endproof

\claim{(3.5) Remark} \rm It should be observed that this formula
cannot be used to solve the equation $\partial u/\partial\ol z=v$ when
$\Supp v$ is not compact; moreover, if $\Supp v$ is compact, a solution
$u$ with  compact support need not always exist.  Indeed, we have a
necessary condition
$$\langle v,z^n\rangle=-\langle u,\partial z^n/\partial\ol z\rangle=0$$
for all integers $n\ge 0$.  Conversely, when the necessary condition
$\langle v,z^n\rangle=0$ is satisfied, the canonical solution $u=(1/\pi
z)\star v$ has compact support: this is easily seen by means of the power
series expansion $(w-z)^{-1}=\sum z^nw^{-n-1}$, if we suppose that 
$\Supp v$ is contained in the disk $|z|<R$ and that $|w|>R$.
\endclaim

\subsection{3.B}{Holomorphic Functions of Several Variables}
Let $\Omega\subset\bC^n$ be an open set. A function $f:\Omega\to\bC$
is said to be holomorphic if $f$ is continuous and separately holomorphic
with respect to each variable, i.e.\ $z_j\mapsto f(\ldots,z_j,\ldots)$
is holomorphic when $z_1,\ldots,z_{j-1}$, $z_{j+1},\ldots,z_n$ are fixed.
The set of holomorphic functions on $\Omega$ is a ring 
and will be denoted $\cO(\Omega)$. We first extend the Cauchy formula
to the case of polydisks. The open polydisk $D(z_0,R)$ of
center $(z_{0,1},\ldots,z_{0,n})$ and (multi)radius $R=(R_1,,\ldots,R_n)$
is defined as the product of the disks of center $z_{0,j}$ and
radius $R_j>0$ in each factor~$\bC$~:
$$D(z_0,R)=D(z_{0,1},R_1)\times\ldots\times D(z_{0,n},R_n)\subset\bC^n.
\leqno(3.6)$$
The {\it distinguished boundary} of $D(z_0,R)$ is by definition
the product of the boundary circles
$$\Gamma(z_0,R)=\Gamma(z_{0,1},R_1)\times\ldots\times\Gamma(z_{0,n},R_n).
\leqno(3.7)$$
It is important to observe that the distinguished boundary is smaller 
than the topological boundary $\partial D(z_0,R)=
\bigcup_j\{z\in\ol D(z_0,R)\,;\,|z_j-z_{0,j}|=R_j\}$ when $n\ge 2$.
By induction on $n$, we easily get the 

\claim{(3.8) Cauchy formula on polydisks} If $\ol D(z_0,R)$ 
is a closed polydisk contained in $\Omega$ and $f\in\cO(\Omega)$, then 
for all $w\in D(z_0,R)$ we have
$$f(w)={1\over(2\pi\ii)^n}\int_{\Gamma(z_0,R)}
{f(z_1,\ldots,z_n)\over(z_1-w_1)\ldots(z_n-w_n)}\,dz_1\ldots dz_n.\eqno\qed$$
\endclaim

The expansion $(z_j-w_j)^{-1}=\sum (w_j-z_{0,j})^{\alpha_j}
(z_j-z_{0,j})^{-\alpha_j-1}$, $\alpha_j\in\bN$, $1\le j\le n$, shows 
that $f$ can be expanded as a convergent power series
$f(w)=\sum_{\alpha\in\bN^n}a_\alpha(w-z_0)^\alpha$ over the polydisk 
$D(z_0,R)$, with the standard notations 
$z^\alpha=z_1^{\alpha_1}\ldots z_n^{\alpha_n}$,
$\alpha!=\alpha_1!\ldots\alpha_n!$ and with
$$a_\alpha=
{1\over(2\pi\ii)^n}\int_{\Gamma(z_0,R)}{f(z_1,\ldots,z_n)\,dz_1\ldots dz_n
\over(z_1-z_{0,1})^{\alpha_1+1}\ldots(z_n-z_{0,n})^{\alpha_n+1}}
={f^{(\alpha)}(z_0)\over\alpha!}.\leqno(3.9)$$
As a consequence, $f$ is holomorphic over $\Omega$ if and only if $f$ is
$\bC$-analytic. Arguments similar to the one variable case easily
yield the

\claim{(3.10) Analytic continuation theorem} If $\Omega$ is
connected and if there exists a point $z_0\in\Omega$ such that
$f^{(\alpha)}(z_0)=0$ for all $\alpha\in\bN^n$, then $f=0$
on~$\Omega$.\qed
\endclaim

Another consequence of (3.9) is the {\it Cauchy inequality}
$$|f^{(\alpha)}(z_0)|\le{\alpha!\over R^\alpha}\sup_{\Gamma(z_0,R)}|f|,~~~~
\ol D(z_0,R)\subset\Omega,\leqno(3.11)$$
From this, it follows that every bounded holomorphic function on $\bC^n$
is constant (Liouville's theorem), and more generally, every holomorphic
function $F$ on $\bC^n$ such that $|F(z)|\le A(1+|z|)^B$ with suitable
constants $A,B\ge 0$ is in fact a polynomial of total degree $\le B$.

We endow $\cO(\Omega)$ with the topology of uniform convergence on
compact sets $K\compact\Omega$, that is, the topology induced by 
$C^0(\Omega,\bC)$. Then $\cO(\Omega)$ is closed in $C^0(\Omega,\bC)$.
The Cauchy inequalities (3.11) show that all derivations $D^\alpha$ are
continuous operators on $\cO(\Omega)$ and that any sequence
$f_j\in\cO(\Omega)$ that is uniformly bounded on  all compact sets
$K\compact\Omega$ is locally equicontinuous. By Ascoli's theorem, we obtain

\claim{(3.12) Montel's theorem} Every locally uniformly bounded
sequence $(f_j)$ in $\cO(\Omega)$ has a convergent subsequence 
$(f_{j(\nu)})$.
\endclaim

In other words, bounded subsets of the Fr\'echet space $\cO(\Omega)$ are
relatively compact (a Fr\'echet space possessing this property is called
a Montel space).

\subsection{3.C}{Differential Calculus on Complex Analytic Manifolds}
A {\it complex analytic manifold} $X$ of dimension $\dim_\bC X=n$
is a differentiable manifold equipped with a holomorphic atlas
$(\tau_\alpha)$ with values in $\bC^n$~; this means by definition
that the transition maps $\tau_{\alpha\beta}$ are holomorphic. The
tangent spaces $T_{X,x}$ then have a natural complex vector space
structure, given by the coordinate isomorphisms
$$d\tau_\alpha(x):T_{X,x}\longrightarrow\bC^n,\qquad U_\alpha\ni x\,;$$
the induced complex structure on $T_{X,x}$ is indeed independent of
$\alpha$ since the differentials $d\tau_{\alpha\beta}$ are $\bC$-linear
isomorphisms. We denote by $T^\bR_X$ the underlying real tangent space
and by  $J\in{\rm End}(T^\bR_X)$ the {\it almost complex structure}, 
i.e.\ the operator of multiplication by 
$\ii=\sqrt{-1}$. If $(z_1,\ldots,z_n)$ are complex analytic coordinates on 
an open subset $\Omega\subset X$ and $z_k=x_k+\ii y_k$, 
then $(x_1,y_1,\ldots,x_n,y_n)$ define real coordinates on $\Omega$, and
$T^\bR_{X\restriction\Omega}$ admits
$(\partial/\partial x_1$, $\partial/\partial y_1$, $\ldots$,
$\partial/\partial x_n$, $\partial/\partial y_n)$ as a basis$\,$;
the almost complex structure is given by
$J(\partial/\partial x_k)=\partial/\partial y_k$,
$J(\partial/\partial y_k)=-\partial/\partial x_k$. The complexified
tangent space $\bC\otimes T_X=\bC\otimes_\bR T^\bR_X=
T^\bR_X\oplus\ii T^\bR_X$ splits into conjugate complex subspaces
which are the eigenspaces of the complexified endomorphism
${\rm Id}\otimes J$ associated to the eigenvalues $\ii$ and $-\ii$. These
subspaces have respective bases
$$~{\partial\over\partial z_k}={1\over 2}\Big({\partial\over\partial x_k}-
\ii{\partial\over\partial y_k}\Big),~~~~
{\partial\over\partial\ol z_k}={1\over 2}\Big({\partial\over\partial x_k}+
\ii{\partial\over\partial y_k}\Big),~~~~1\le k\le n\leqno(3.13)$$
and are denoted $T^{1,0}X$ ({\it holomorphic vectors} or {\it 
vectors of type} $(1,0)$) and $T^{0,1}X$ ({\it antiholomorphic vectors} or
{\it vectors of type} $(0,1)$). The subspaces $T^{1,0}X$ and $T^{0,1}X$
are canonically isomorphic to the complex tangent space 
$T_X$ (with complex structure $J$) and its conjugate $\ol{T_X}$ 
(with conjugate complex structure $-J$), via the $\bC$-linear embeddings
$$\cmalign{T_X&\longrightarrow T^{1,0}_X\subset\bC\otimes T_X,
\hfill\ol{T_X}&\longrightarrow T^{0,1}_X\subset\bC\otimes T_X\cr
\hfill\xi&\longmapsto{1\over2}(\xi-\ii J\xi),~~~~~
\xi&\longmapsto{1\over2}(\xi+\ii J\xi).\cr}$$
We thus have a canonical decomposition $\bC\otimes T_X=T^{1,0}_X\oplus
T^{0,1}_X\simeq T_X\oplus\ol{T_X}$, and by duality a decomposition
$$\Hom_\bR(T^\bR_X;\bC)\simeq\Hom_\bC(\bC\otimes T_X;\bC)\simeq
T^\star_X\oplus\ol{T^\star_X}$$
where $T^\star_X$ is the space of $\bC$-linear forms and 
$\ol{T^\star_X}$ the space of conjugate $\bC$-linear forms.
With these notations, $(dx_k,dy_k)$ is a basis of $\Hom_\bR(T_\bR
X,\bC)$, $(dz_j)$ a basis of $T^\star_X$, $(d\ol z_j)$ a basis of
$\ol{T^\star_X}$, and the differential of a function $f\in C^1(\Omega,\bC)$
can be written
$$df=\sum_{k=1}^n
{\partial f\over\partial x_k}\,dx_k+{\partial f\over\partial y_k}\,dy_k
=\sum_{k=1}^n
{\partial f\over\partial z_k}\,dz_k+{\partial f\over\partial\ol z_k}\,
d\ol z_k.\leqno(3.14)$$
The function $f$ is holomorphic on $\Omega$ if and only if $df$ is 
$\bC$-linear, i.e.\ if and only if $f$ satisfies the
{\it Cauchy-Riemann equations} $\partial f/\partial\ol z_k=0$ on
$\Omega$, $1\le k\le n$. We still denote here by $\cO(X)$ the algebra
of holomorphic functions on $X$.
 
Now, we study the basic rules of complex differential calculus.
The complexified exterior algebra
$\bC\otimes_\bR\Lambda^\bu_\bR(T_X^\bR)^\star
=\Lambda^\bu_\bC(\bC\otimes T_X)^\star$ is given by
$$\Lambda^k(\bC\otimes T_X)^\star=\Lambda^k\big(T_X\oplus\ol{T_X}\big)^\star
=\bigoplus_{p+q=k}\Lambda^{p,q}T^\star_X,~~~~0\le k\le 2n$$
where the exterior products are taken over $\bC$, and where the components
$\Lambda^{p,q}T^\star_X$ are defined by
$$\Lambda^{p,q}T^\star_X=\Lambda^p T^\star_X\otimes
\Lambda^q\ol{T^\star_X}.\leqno(3.15)$$
A complex differential form $u$ on $X$ is said to be of {\it bidegree} or
{\it type} $(p,q)$ if its value at every point lies in the component 
$\Lambda^{p,q}T^\star_X$~; we shall denote by $C^s(\Omega,\Lambda^{p,q}
T_X^\star)$ the space of differential forms of bidegree $(p,q)$ and class
$C^s$ on any open  subset $\Omega$ of $X$. If $\Omega$ is a coordinate
open set, such a form can be written
$$u(z)=\sum_{|I|=p,|J|=q}u_{I,J}(z)\,dz_I\wedge d\ol z_J,~~~~
u_{I,J}\in C^s(\Omega,\bC).$$
This writing is usually much more convenient than the expression in
terms of the real basis $(dx_I\wedge dy_J)_{|I|+|J|=k}$ which is not
compatible with the splitting of $\Lambda^k T^\star_\bC X$ in its
$(p,q)$ components. Formula (3.14) shows that the exterior derivative 
$d$ splits into $d=d'+d''$, where
$$\leqalignno{
d'&:\cC^\infty(X,\Lambda^{p,q}T^\star_X)\longrightarrow \cC^\infty(X,\Lambda^{p+1,q}T^\star_X),\cr
d''&:\cC^\infty(X,\Lambda^{p,q}T^\star_X)\longrightarrow \cC^\infty(X,\Lambda^{p,q+1}T^\star_X),\cr
d'u&=\sum_{I,J}\sum_{1\le k\le n}{\partial u_{I,J}\over\partial z_k}\,dz_k
\wedge dz_I\wedge d\ol z_J,&(3.16')\cr
d''u&=\sum_{I,J}\sum_{1\le k\le n.}{\partial u_{I,J}\over\partial\ol z_k}
\,d\ol z_k\wedge dz_I\wedge d\ol z_J.&(3.16'')\cr}$$
The identity $d^2=(d'+d'')^2=0$ is equivalent to
$$d^{\prime 2} = 0,~~~~d'd''+d''d' = 0,~~~~d^{\prime\prime 2}=0,
\leqno(3.17)$$
since these three operators send $(p,q)$-forms in $(p+2,q)$, $(p+1,q+1)$ and
$(p,q+2)$-forms, respectively. In particular, the operator $d''$ defines for
each $p=0,1,\ldots,n$ a complex, called the {\it Dolbeault complex}
$$\cC^\infty(X,\Lambda^{p,0}T^\star_X)\buildo d''\over\longrightarrow\cdots\longrightarrow
\cC^\infty(X,\Lambda^{p,q}T^\star_X)\buildo d''\over\longrightarrow
\cC^\infty(X,\Lambda^{p,q+1}T^\star_X)$$
and corresponding {\it Dolbeault cohomology groups}
$$H^{p,q}(X,\bC)={\Ker d^{\prime\prime\,p,q}\over
\Im d^{\prime\prime\,p,q-1}},\leqno(3.18)$$
with the convention that the image of $d''$ is zero for $q=0$. The
coho\-mo\-logy group $H^{p,0}(X,\bC)$ consists of $(p,0)$-forms
$u=\sum_{|I|=p}u_I(z)\,dz_I$ such that $\partial u_I/\partial\ol z_k=0$
for all $I,k$, i.e.\ such that all coefficients $u_I$ are holomorphic.
Such a form is called a {\it holomorphic $p$-form} on $X$.

Let $F:X_1\longrightarrow X_2$ be a holomorphic map between complex manifolds. The
pull-back $F^\star u$ of a $(p,q)$-form $u$ of bidegree $(p,q)$ on $X_2$
is again homogeneous of bidegree $(p,q)$, because the components
$F_k$ of $F$ in any coordinate chart are holomorphic, hence
$F^\star dz_k=dF_k$ is \hbox{$\bC$-linear.} In particular, the equality
$dF^\star u=F^\star du$ implies
$$d'F^\star u=F^\star d'u,~~~~d''F^\star u=F^\star d''u.
\leqno(3.19)$$
Note that these commutation relations are no longer true for a non
holomorphic change of variable. As in the case of the De Rham cohomology
groups, we get a pull-back morphism
$$F^\star: H^{p,q}(X_2,\bC)\longrightarrow H^{p,q}(X_1,\bC).$$
The rules of complex differential calculus can be easily extended to currents.
We use the following notation.

\claim{(3.20) Definition} There are decompositions
$$\cD^k(X,\bC)=\bigoplus_{p+q=k}\cD^{p,q}(X,\bC),~~~~
\cD'_k(X,\bC)=\bigoplus_{p+q=k}\cD'_{p,q}(X,\bC).$$
The space $\cD'_{p,q}(X,\bC)$ is called the space of currents of
bidimension $(p,q)$ and bidegree $(n-p,n-q)$ on~$X$, and is also
denoted $\cD^{\prime\,n-p,n-q}(X,\bC)$.
\endclaim

\subsection{3.D}{Newton and Bochner-Martinelli Kernels}
The {\it Newton kernel} is the elementary solution of the usual Laplace
operator $\Delta=\sum\partial^2/\partial x_j^2$ in~$\bR^m$.
We first recall a construction of the Newton kernel.

Let $d\lambda=dx_1\ldots dx_m$ be the Lebesgue measure on $\bR^m$. 
We denote by $B(a,r)$ the euclidean open ball of 
center $a$ and radius $r$ in $\bR^m$ and by $S(a,r)=\partial B(a,r)$ 
the corresponding sphere. Finally, we set $\alpha_m=\Vol\big(B(0,1)\big)$
and $\sigma_{m-1}=m\alpha_m$ so that
$$\Vol\big(B(a,r)\big)=\alpha_mr^m,~~~~{\rm Area}\big(S(a,r)\big)=
\sigma_{m-1}r^{m-1}.\leqno(3.21)$$
The second equality follows from the first by derivation. An explicit 
computation  of the integral $\int_{\bR^m}e^{-|x|^2}d\lambda(x)$ in polar
coordinates shows that $\alpha_m=\pi^{m/2}/(m/2)!$ where $x!=\Gamma(x+1)$
is the Euler Gamma function. The {\it Newton kernel} is then given by:
$$\cases{
N(x)=\displaystyle{1\over 2\pi}\,\log|x|&if~~$m=2$,\cr
N(x)=\displaystyle-{1\over(m-2)\sigma_{m-1}}\,|x|^{2-m}
&if~~$m\ne 2$.\cr}\leqno(3.22)$$
The function $N(x)$ is locally integrable on $\bR^m$ and satisfies
$\Delta N=\delta_0$. When $m=2$, this follows from Cor.~3.4
and the fact that $\Delta=4\partial^2/\partial z\partial\ol z$. 
When $m\ne 2$, this can be checked by computing the weak limit
$$\eqalign{
\lim_{\varepsilon\to 0}~\Delta(|x|^2+\varepsilon^2)^{1-m/2}&=
\lim_{\varepsilon\to 0}~m(2-m)\varepsilon^2(|x|^2+\varepsilon^2)^{-1-m/2}\cr
&=m(2-m)\,I_m\,\delta_0\cr}$$
with~ $I_m=\int_{\bR^m}(|x|^2+1)^{-1-m/2}\,d\lambda(x)$. The last
equality is easily seen by performing the change of variable $y=\varepsilon x$
in the integral 
$$\int_{\bR^m}\varepsilon^2(|x|^2+\varepsilon^2)^{-1-m/2}
\,f(x)\,d\lambda(x)=\int_{\bR^m}(|y|^2+1)^{-1-m/2}
\,f(\varepsilon y)\,d\lambda(y),$$
where $f$ is an arbitrary test function. Using polar coordinates, we find
that $I_m=\sigma_{m-1}/m$ and our formula follows.

The {\it Bochner-Martinelli kernel} is the $(n,n-1)$-differential
form on $\bC^n$ with $L^1_\loc$ coefficients defined by
$$\leqalignno{
k_\BM(z)&=c_n\sum_{1\le j\le n}(-1)^j
{\ol z_j\,dz_1\wedge\ldots dz_n\wedge d\ol z_1\wedge\ldots
\widehat{d\ol z_j}\ldots\wedge d\ol z_n\over|z|^{2n}},&(3.23)\cr
c_n&=(-1)^{n(n-1)/2}{(n-1)!\over(2\pi\ii)^n}.\cr}$$

\claim{(3.24) Lemma} $d''k_\BM=\delta_0$ on $\bC^n$.
\endclaim

\proof{Proof} Since the Lebesgue measure on $\bC^n$ is
$$d\lambda(z)=\bigwedge_{1\le j\le n}{\ii\over 2}dz_j\wedge d\ol z_j
=\Big({\ii\over 2}\Big)^n(-1)^{\textstyle{n(n-1)\over 2}}
dz_1\wedge\ldots dz_n\wedge d\ol z_1\wedge\ldots d\ol z_n,$$
we find
$$\eqalignno{
d''k_\BM&=-{(n-1)!\over\pi^n}\sum_{1\le j\le n}{\partial\over\partial
\ol z_j}\Big({\ol z_j\over |z|^{2n}}\Big)d\lambda(z)\cr
&=-{1\over n(n-1)\alpha_{2n}}\sum_{1\le j\le n}{\partial^2\over\partial
z_j\partial\ol z_j}\Big({1\over |z|^{2n-2}}\Big)d\lambda(z)\cr
&=\Delta N(z)d\lambda(z)=\delta_0.&\square\cr}$$
\endproof

We let $K_\BM(z,\zeta)$ be the pull-back of $k_\BM$ by the
map $\pi:\bC^n\times\bC^n\to\bC^n$, $(z,\zeta)\longmapsto z-\zeta$.
Then Formula (2.19) implies
$$d''K_\BM=\pi^\star\delta_0=[\Delta],\leqno(3.25)$$
where $[\Delta]$ denotes the current of integration on the diagonal
$\Delta\subset\bC^n\times\bC^n$.

\claim{(3.26) Koppelman formula} Let $\Omega\subset\bC^n$ be a bounded
open set with piecewise $C^1$ boundary. Then for every $(p,q)$-form $v$ of
class $C^1$ on $\ol\Omega$ we have
$$\eqalign{
v(z)=\int_{\partial\Omega}&K_\BM^{p,q}(z,\zeta)\wedge v(\zeta)\cr
&\quad{}+d''_z\int_\Omega K_\BM^{p,q-1}(z,\zeta)\wedge v(\zeta)+
\int_\Omega K_\BM^{p,q}(z,\zeta)\wedge d''v(\zeta)\cr}$$
on $\Omega$, where
$K_\BM^{p,q}(z,\zeta)$ denotes the component of $K_\BM(z,\zeta)$
of type $(p,q)$ in~$z$ and $(n-p,n-q-1)$ in~$\zeta$.
\endclaim

\proof{Proof} Given $w\in\cD^{n-p,n-q}(\Omega)$, we consider the integral
$$\int_{\partial\Omega\times\Omega}K_\BM(z,\zeta)\wedge v(\zeta)\wedge
w(z).$$
It is well defined since $K_\BM$ has no singularities on
$\partial\Omega \times\Supp v\compact\partial\Omega\times\Omega$. Since
$w(z)$ vanishes on $\partial\Omega$ the integral can be extended as
well to $\partial(\Omega\times \Omega)$. As $K_\BM(z,\zeta)\wedge
v(\zeta)\wedge w(z)$ is of total bidegree $(2n,2n-1)$, its differential
$d'$ vanishes. Hence Stokes' formula yields
$$\eqalign{
\int_{\partial\Omega\times\Omega}&K_\BM(z,\zeta)\wedge v(\zeta)\wedge
w(z)=\int_{\Omega\times\Omega}d''\big(K_\BM(z,\zeta)\wedge v(\zeta)\wedge
w(z)\big)\cr
&=\int_{\Omega\times\Omega}d''K_\BM(z,\zeta)\wedge v(\zeta)\wedge w(z)
-K^{p,q}_\BM(z,\zeta)\wedge d''v(\zeta)\wedge w(z)\cr
&\qquad-(-1)^{p+q}\int_{\Omega\times\Omega}K^{p,q-1}_\BM(z,\zeta)\wedge
v(\zeta)\wedge d''w(z).\cr}$$
By (3.25) we have
$$\int_{\Omega\times\Omega}\!\!\!d''K_\BM(z,\zeta)\wedge v(\zeta)\wedge
w(z)=\int_{\Omega\times\Omega}\![\Delta]\wedge v(\zeta)\wedge
w(z)=\int_\Omega v(z)\wedge w(z)$$
Denoting $\langle~,~\rangle$ the pairing between currents and test forms
on $\Omega$, the above equality is thus equivalent to
$$\eqalign{
\langle\int_{\partial\Omega}K_\BM(z,\zeta)\wedge v(\zeta),w(z)\rangle
&=\langle v(z)-\int_\Omega K^{p,q}_\BM(z,\zeta)\wedge d''v(\zeta),w(z)
\rangle\cr
&{}-(-1)^{p+q}\langle\int_\Omega K^{p,q-1}_\BM(z,\zeta)\wedge
v(\zeta),d''w(z)\rangle,\cr}$$
which is itself equivalent to the Koppelman formula by integrating
$d''v$ by parts.\qed
\endproof

\claim{(3.27) Corollary} Let $v\in{}^s\cD^{p,q}(\bC^n)$ be a form
of class $C^s$ with compact support such that $d''v=0$, $q\ge 1$.
Then the $(p,q-1)$-form
$$u(z)=\int_{\bC^n}K_\BM^{p,q-1}(z,\zeta)\wedge v(\zeta)$$
is a $C^s$ solution of the equation $d''u=v$. Moreover, if $(p,q)=(0,1)$
and $n\ge 2$ then $u$ has compact support, thus the Dolbeault cohomology
group with compact support $H^{0,1}_c(\bC^n,\bC)$ vanishes for
$n\ge 2$.
\endclaim

\proof{Proof} Apply the Koppelman formula on a sufficiently large ball
$\ol\Omega=\ol B(0,R)$ containing $\Supp v$. Then the formula
immediately gives $d''u=v$. Observe that the coefficients of
$K_\BM(z,\zeta)$ are $O(|z-\zeta|^{-(2n-1)})$, hence
$|u(z)|=O(|z|^{-(2n-1)})$ at infinity. If $q=1$, then $u$ is holomorphic
on $\bC^n\ssm\ol B(0,R)$. Now, this complement is a union of complex
lines when $n\ge 2$, hence $u=0$ on $\bC^n\ssm\ol B(0,R)$ by
Liouville's theorem.\qed
\endproof

\claim{(3.28) Hartogs extension theorem} Let $\Omega$ be an open set in
$\bC^n$, $n\ge 2$, and let $K\subset\Omega$ be a compact subset such
that $\Omega\ssm K$ is connected. Then every holomorphic function
$f\in\cO(\Omega\ssm K)$ extends into a function $\wt f\in\cO(\Omega)$.
\endclaim

\proof{Proof} Let $\psi\in\cD(\Omega)$ be a cut-off function equal to
$1$ on a neighborhood of~$K$. Set $f_0=(1-\psi)f\in \cC^\infty(\Omega)$,
defined as $0$ on~$K$. Then $v=d''f_0=-fd''\psi$ can be extended by $0$
outside $\Omega$, and can thus be seen as a smooth $(0,1)$-form with
compact support in~$\bC^n$, such that $d''v=0$. By Cor.~3.27, there is
a smooth function $u$ with compact support in $\bC^n$ such that $d''u=v$.
Then $\wt f=f_0-u\in\cO(\Omega)$. Now $u$ is holomorphic outside $\Supp\psi$,
so $u$ vanishes on the unbounded component $G$ of $\bC^n\ssm\Supp\psi$.
The boundary $\partial G$ is contained in $\partial\Supp\psi\subset
\Omega\ssm K$, so $\wt f=(1-\psi)f-u$ coincides with $f$ on the non empty
open set $\Omega\cap G\subset\Omega\ssm K$. Therefore $\wt f=f$ on
the connected open set $\Omega\ssm K$.\qed
\endproof

A refined version of the Hartogs extension theorem due to Bochner will
be given in Exercise~8.13. It shows that $f$ need only be given as a $C^1$
function on $\partial\Omega$, satisfying the tangential Cauchy-Riemann
equations (a so-called {\it CR-function}). Then $f$ extends as a holomorphic
function $\wt f\in\cO(\Omega)\cap C^0(\ol\Omega)$, provided that
$\partial\Omega$ is connected. 

\subsection{3.E}{The Dolbeault-Grothendieck Lemma}
We are now in a position to prove the Dolbeault-Grothendieck lemma
\cite{Dolbeault 1953}, which is the analogue for $d''$ of the Poincar\'e
lemma. The proof given below makes use of the Bochner-Martinelli kernel.
Many other proofs can be given, e.g.\ by using a reduction to the one
dimensional case in combination with the Cauchy formula~(3.2),
see Exercise~8.5 or \cite{H\"ormander 1966}.

\claim{(3.29) Dolbeault-Grothendieck lemma} Let $\Omega$ 
be a neighborhood of $0$ in $\bC^n$ and $v\in{}^s\cE^{p,q}(\Omega,\bC)$,
$[$resp.\ $v\in{}^s\cD^{\prime\,p,q}(\Omega,\bC)]$, such that $d''v=0$,
where $1\le s\le\infty$.
\medskip
\item{\rm a)} If $q=0$, then $v(z)=\sum_{|I|=p}v_I(z)\,dz_I$ is a holomorphic
$p$-form, i.e.\ a form whose coefficients are holomorphic functions.
\smallskip
\item{\rm b)} If $q\ge 1$, there exists a neighborhood $\omega\subset
\Omega$ of $0$ and a form $u$ in ${}^s\cE^{p,q-1}(\omega,\bC)$
$[$resp.\ a current $u\in{}^s\cD^{\prime\,p,q-1}(\smash{\omega},\bC)]$
such that $d''u=v$ on~$\omega$.
\vskip0pt
\endclaim

\proof{Proof} We assume that $\Omega$ is a ball $B(0,r)\subset\bC^n$
and take for simplicity $r>1$ (possibly after a dilation of coordinates).
We then set $\omega=B(0,1)$. Let $\psi\in\cD(\Omega)$
be a cut-off function equal to $1$ on $\omega$. The Koppelman formula
(3.26) applied to the form $\psi v$ on $\Omega$ gives
$$\psi(z)v(z)=d''_z\!\int_\Omega\! K_\BM^{p,q-1}(z,\zeta)\wedge
\psi(\zeta)v(\zeta)+{}\!\int_\Omega\! K_\BM^{p,q}(z,\zeta)\wedge
d''\psi(\zeta)\wedge v(\zeta).$$
This formula is valid even when $v$ is a current, because we may
regularize $v$ as $v\star\rho_\varepsilon$ and take the limit.
We introduce on $\bC^n\times\bC^n\times\bC^n$ the kernel
$$K(z,w,\zeta)=c_n\sum_{j=1}^n{(-1)^j(w_j-\ol\zeta_j)
\over((z-\zeta)\cdot(w-\ol\zeta))^n}
\bigwedge\limits_k(dz_k-d\zeta_k)\wedge
\bigwedge\limits_{k\ne j}(dw_k-d\ol\zeta_k).$$
By construction, $K_\BM(z,\zeta)$ is the result of the substitution
$w=\ol z$ in $K(z,w,\zeta)$, i.e.\ $K_\BM=h^\star K$ where
$h(z,\zeta)=(z,\ol z,\zeta)$. We denote by $K^{p,q}$ the component
of $K$ of bidegree $(p,0)$ in $z$, $(q,0)$ in $w$ and $(n-p,n-q-1)$
in~$\zeta$. Then $K_\BM^{p,q}=h^\star K^{p,q}$ and we find
$$v=d''u_0+g^\star v_1~~~~\hbox{\rm on $\omega$,}$$
where $g(z)=(z,\ol z)$ and
$$\eqalign{
u_0(z)&=\int_\Omega K_\BM^{p,q-1}(z,\zeta)\wedge\psi(\zeta)v(\zeta),\cr
v_1(z,w)&=\int_\Omega K^{p,q}(z,w,\zeta)\wedge d''\psi(\zeta)\wedge v(\zeta).
\cr}$$
By definition of $K^{p,q}(z,w,\zeta)$, $v_1$ is holomorphic on the open set
$$U=\big\{(z,w)\in\omega\times\omega\,;\,\forall\zeta\notin\omega,~
\Re(z-\zeta)\cdot(w-\ol\zeta)>0\big\},$$
which contains the ``conjugate-diagonal'' points $(z,\ol z)$
as well as the points $(z,0)$ and $(0,w)$ in $\omega\times\omega$.
Moreover $U$ clearly has convex slices $(\{z\}\times\bC^n)\cap U$
and $(\bC^n\times\{w\})\cap U$. In particular $U$ is starshaped with
respect to $w$, i.e.\
$$(z,w)\in U\Longrightarrow (z,tw)\in U,~~~~\forall t\in[0,1].$$
As $u_1$ is of type $(p,0)$ in $z$ and $(q,0)$ in $w$, we get
$d''_z(g^\star v_1)=g^\star d_wv_1=0$, hence $d_wv_1=0$.
For~$q=0$ we have $K_\BM^{p,q-1}=0$, thus $u_0=0$, and $v_1$ does
not depend on $w$, thus $v$ is holomorphic on~$\omega$. For
$q\ge 1$, we can use the homotopy formula (1.23) with respect to
$w$ (considering $z$ as a parameter) to get a holomorphic form
$u_1(z,w)$ of type $(p,0)$ in $z$ and $(q-1,0)$ in~$w$, such
that $d_wu_1(z,w)=v_1(z,w)$. Then we get $d''g^\star u_1=
g^\star d_wu_1=g^\star v_1$, hence
$$v=d''(u_0+g^\star u_1)~~~~\hbox{\rm on $\omega$}.$$
Finally, the coefficients of $u_0$ are obtained as linear combinations of
convolutions of the coefficients of $\psi v$ with $L^1_\loc$ functions
of the form $\ol\zeta_j|\zeta|^{-2n}$. Hence $u_0$ is of class $C^s$
(resp.\ is a current of order $s$), if $v$ is.\qed
\endproof

\claim{(3.30) Corollary} The operator $d''$ is hypoelliptic in bidegree
$(p,0)$, i.e.~if a current $f\in\cD^{\prime\,p,0}(X,\bC)$
satisfies $d''f\in\cE^{p,1}(X,\bC)$, then $f\in\cE^{p,0}(X,\bC)$.
\endclaim

\proof{Proof} The result is local, so we may assume that $X=\Omega$ is a
neighborhood of $0$ in $\bC^n$. The $(p,1)$-form $v=d''f\in
\cE^{p,1}(X,\bC)$ satisfies $d''v=0$, hence there exists 
$u\in\cE^{p,0}(\smash{\wt\Omega},\bC)$ such that
$d''u=d''f$. Then $f-u$ is holomorphic and $f=(f-u)+u\in\cE^{p,0}
(\smash{\wt\Omega},\bC)$.\qed
\endproof

\section{4}{Subharmonic Functions}
A {\it harmonic} (resp.\ {\it subharmonic}) function on an open subset
of $\bR^m$ is essentially a function (or distribution) $u$ such that
$\Delta u=0$ (resp.\ $\Delta u\ge 0$). A fundamental example of
subharmonic function is given by the Newton kernel~$N$, which is
actually harmonic on $\bR^m\ssm\{0\}$. Subharmonic functions are an
essential tool of harmonic analysis and potential theory. Before giving
their precise definition and properties, we derive a basic integral
formula involving the Green kernel of the Laplace operator on the ball.

\subsection{4.A}{Construction of the Green Kernel} 
The {\it Green kernel} $G_\Omega(x,y)$ of a smoothly bounded domain
$\Omega\compact\bR^m$ is the solution of the following {\it Dirichlet
boundary problem} for the Laplace operator $\Delta$ on~$\Omega\,$:

\claim{(4.1) Definition} The Green kernel of a smoothly bounded domain
$\Omega\compact\bR^m$ is a function $G_\Omega(x,y):\ol\Omega\times
\ol\Omega\to[-\infty,0]$ with the following properties:
\medskip
\item{\rm a)} $G_\Omega(x,y)$ is $\cC^\infty$ on $\ol\Omega\times\ol\Omega
\ssm\Diag_\Omega$~~ $(\Diag_\Omega={}$ diagonal$\,)~;$
\smallskip
\item{\rm b)} $G_\Omega(x,y)=G_\Omega(y,x)~;$
\smallskip
\item{\rm c)} $G_\Omega(x,y)<0$ on $\Omega\times\Omega$ and $G_\Omega(x,y)=0$
on $\partial\Omega\times\Omega\,;$
\smallskip
\item{\rm d)} $\Delta_x G_\Omega(x,y)=\delta_y$~ on $\Omega$ 
for every fixed $y\in\Omega$.
\endclaim

It can be shown that $G_\Omega$ always exists and is unique. The
uniqueness is an easy consequence of the maximum principle
(see Th.~4.14 below). In the case where $\Omega=B(0,r)$ is a ball (the
only case we are going to deal with), the existence can be shown through
explicit calculations. In fact the Green kernel $G_r(x,y)$ of $B(0,r)$ is
$$G_r(x,y)=N(x-y)-N\Big({|y|\over r}\Big(x-{r^2\over|y|^2}\,y\Big)\Big),~~~~
x,y\in\ol B(0,r).\leqno(4.2)$$
A substitution of the explicit value of $N(x)$ yields:
$$\eqalign{
&G_r(x,y)={1\over 4\pi}\log{|x-y|^2\over r^2-2\langle x,y\rangle+{1\over r^2}
|x|^2\,|y|^2}~~~~\hbox{\rm if}~~m=2,~~~\hbox{\rm otherwise}\cr
&G_r(x,y)={-1\over(m-2)\sigma_{m-1}}\Big(|x-y|^{2-m}-\big(r^2-
2\langle x,y\rangle+{1\over r^2}|x|^2\,|y|^2\big)^{1-m/2}\Big).\cr}$$

\claim{(4.3) Theorem} The above defined function $G_r$ satisfies all four
properties $(4.1\,\hbox{\rm a--d})$ on $\Omega=B(0,r)$, thus $G_r$ is
the Green kernel of $B(0,r)$.
\endclaim

\proof{Proof} The first three properties are immediately verified on the 
formulas, because
$$r^2-2\langle x,y\rangle+{1\over r^2}|x|^2\,|y|^2=|x-y|^2+{1\over r^2}
\big(r^2-|x|^2\big)\big(r^2-|y|^2\big).$$
For property d), observe that $r^2y/|y|^2\notin\ol B(0,r)$ whenever
$y\in B(0,r)\ssm\{0\}$. The second Newton kernel in the right hand
side of (4.1) is thus harmonic in $x$ on $B(0,r)$, and
$$\Delta_xG_r(x,y)=\Delta_x N(x-y)=\delta_y~~~\hbox{\rm 
on}~~B(0,r).\eqno\qed$$
\endproof

\subsection{4.B}{Green-Riesz Representation Formula and Dirichlet Problem}
\subsubsection{4.B.1}{Green-Riesz Formula.}
For all smooth functions $u,v$ on a smoothly bounded domain $\Omega\compact
\bR^m$, we have
$$\int_\Omega (u\,\Delta v-v\,\Delta u)\,d\lambda=\int_{\partial\Omega}
\Big(u\,{\partial v\over\partial\nu}-v\,{\partial u\over\partial\nu}\Big)\,
d\sigma\leqno(4.4)$$
where $\partial/\partial\nu$ is the derivative along the outward normal
unit vector $\nu$ of $\partial\Omega$ and $d\sigma$ the euclidean area measure.
Indeed
$$(-1)^{j-1}\,dx_1\wedge\ldots\wedge\wh{dx_j}\wedge\ldots
\wedge dx_{m\,\restriction\partial\Omega}=\nu_j\,d\sigma,$$
for the wedge product of $\langle\nu,dx\rangle$ with the left hand side 
is $\nu_j\,d\lambda$. Therefore
$${\partial v\over\partial\nu}\,d\sigma
=\sum_{j=1}^m~{\partial v\over\partial x_j}\,\nu_j\,d\sigma
=\sum_{j=1}^m~(-1)^{j-1}{\partial v
\over\partial x_j}\,dx_1\wedge\ldots\wedge\wh{dx_j}\wedge\ldots\wedge dx_m.$$
Formula (4.4) is then an easy consequence of Stokes' theorem. Observe that
(4.4) is still valid if $v$ is a distribution with singular support
relatively compact in $\Omega$. For $\Omega=B(0,r)$, $u\in C^2\big(\ol 
B(0,r),\bR\big)$ and $v(y)=G_r(x,y)$, we get the {\it Green-Riesz
representation formula}\/:
$$u(x)=\int_{B(0,r)}\Delta u(y)\,G_r(x,y)\,d\lambda(y)+\int_{S(0,r)}u(y)\,P_r(x,y)
\,d\sigma(y)\leqno(4.5)$$
where $P_r(x,y)=\partial G_r(x,y)/\partial\nu(y)$, $(x,y)\in B(0,r)\times
S(0,r)$. The function $P_r(x,y)$ is called the {\it Poisson kernel}. 
It is smooth and satisfies $\Delta_xP_r(x,y)=0$ on $B(0,r)$ by (4.1~d).
A simple computation left to the reader yields:
$$P_r(x,y)={1\over\sigma_{m-1}r}\,{r^2-|x|^2\over|x-y|^m}.\leqno(4.6)$$
Formula (4.5) for $u\equiv 1$ shows that $\smash{\int_{S(0,r)}}P_r(x,y)\,
d\sigma(y)=1$. When $x$ in $B(0,r)$ tends to $x_0\in S(0,r)$, we see
that $P_r(x,y)$ converges uniformly to $0$ on every compact subset of
$S(0,r)\ssm\{x_0\}$~; it follows that the measure 
$P_r(x,y)\,d\sigma(y)$ converges weakly to $\delta_{x_0}$ on $S(0,r)$.

\subsubsection{4.B.2}{Solution of the Dirichlet Problem.} 
For any bounded measurable function $v$ on $S(a,r)$ we define
$$P_{a,r}[v](x)=\int_{S(a,r)}v(y)\,P_r(x-a,y-a)\,d\sigma(y),~~~~
x\in B(a,r).\leqno(4.7)$$
If $u\in C^0\big(\ol B(a,r),\bR\big)\cap C^2\big(B(a,r),\bR\big)$ is
harmonic, i.e.\ $\Delta u=0$ on $B(a,r)$, then (4.5) gives $u=P_{a,r}[u]$
on $B(a,r)$, i.e.\ the Poisson kernel reproduces harmonic functions.
Suppose now that $v\in C^0\big(S(a,r),\bR\big)$ is given.
Then $P_r(x-a,y-a)\,d\sigma(y)$ converges weakly to $\delta_{x_0}$ when
$x$ tends to $x_0\in S(a,r)$, so $P_{a,r}[v](x)$ converges to $v(x_0)$.
It follows that the function $u$ defined by
$$\cases{u=P_{a,r}[v]&on~~$B(a,r)$,\cr
         u=v         &on~~$S(a,r)$\cr}$$
is continuous on $\ol B(a,r)$ and harmonic on $B(a,r)$~;
thus $u$ is the solution of the Dirichlet problem with boundary values $v$.

\subsection{4.C}{Definition and Basic Properties of Subharmonic Functions}
\subsubsection{4.C.1}{Definition. Mean Value Inequalities.}
If $u$ is a Borel function on $\ol B(a,r)$ which is
bounded above or below, we consider the mean values of $u$ over the ball
or sphere:
$$\leqalignno{
\mu_B(u\,;a,r)&={1\over\alpha_m r^m}\int_{B(a,r)}u(x)\,d\lambda(x),&(4.8)\cr
\mu_S(u\,;a,r)&={1\over\sigma_{m-1} r^{m-1}}\int_{S(a,r)}u(x)\,d\sigma(x).
&(4.8')\cr}$$
As $d\lambda=dr\,d\sigma$ these mean values are related by
$$\leqalignno{
\mu_B(u\,;a,r)&={1\over\alpha_mr^m}\int_0^r\sigma_{m-1}t^{m-1}\,
\mu_S(u\,;a,t)\,dt&(4.9)\cr
&=m\int_0^1 t^{m-1}\,\mu_S(u\,;a,rt)\,dt.\cr}$$
Now, apply formula (4.5) with $x=0$. We get 
$P_r(0,y)=1/\sigma_{m-1}r^{m-1}$ and 
$G_r(0,y)=(|y|^{2-m}-r^{2-m})/(2-m)\sigma_{m-1}=
-(1/\sigma_{m-1})\int^r_{|y|}t^{1-m}dt$, thus
$$\eqalign{
\int_{B(0,r)}\Delta u(y)\,G_r(0,y)\,d\lambda(y)&=-{1\over\sigma_{m-1}}\int_0^r
{dt\over t^{m-1}}\int_{|y|<t}\Delta u(y)\,d\lambda(y)\cr
&=-{1\over m}\int_0^r\mu_B(\Delta u\,;0,t)\,t\,dt\cr}$$
thanks to the Fubini formula. By translating $S(0,r)$ to $S(a,r)$, (4.5)
implies the {\it Gauss formula}
$$\mu_S(u\,;a,r)=u(a)+{1\over m}\int_0^r\mu_B(\Delta u\,;a,t)\,t\,dt.
\leqno(4.10)$$
Let $\Omega$ be an open subset of $\bR^m$ and $u\in C^2(\Omega,\bR)$. If
$a\in\Omega$ and \hbox{$\Delta u(a)>0$} (resp.\ $\Delta u(a)<0$),
Formula (4.10) shows that $\mu_S(u\,;a,r)>u(a)$ (resp.\
$\mu_S(u\,;a,r)<u(a)$) for $r$ small enough. In particular, $u$ is harmonic
(i.e.\ $\Delta u=0$) if and only if $u$ satisfies the {\it mean value equality}
$$\mu_S(u\,;a,r)=u(a),~~~~\forall\ol B(a,r)\subset\Omega.$$
Now, observe that if $(\rho_\varepsilon)$ is a family of radially symmetric
smoothing kernels associated with $\rho(x)=\wt\rho(|x|)$ and if
$u$ is a Borel locally bounded function, an easy computation yields
$$\leqalignno{u\star\rho_\varepsilon(a)
&=\int_{B(0,1)}u(a+\varepsilon x)\,\rho(x)\,d\lambda\cr
&=\sigma_{m-1}\int_0^1\mu_S(u\,;a,\varepsilon t)\,\wt\rho(t)\,t^{m-1}\,dt.
&(4.11)\cr}$$
Thus, if $u$ is a Borel locally bounded function satisfying the mean value
equality on~$\Omega$, (4.11) shows that $u\star\rho_\varepsilon=u$ on
$\Omega_\varepsilon$, in particular $u$ must be smooth. Similarly, if we
replace the mean value equality by an inequality, the relevant regularity
property to be required for $u$ is just semicontinuity.

\claim{(4.12) Theorem and definition} Let 
$u:\Omega\longrightarrow[-\infty,+\infty[$ be an upper semicontinuous function.
The following various forms of mean value inequalities are equivalent:
\medskip
\item{\rm a)} $u(x)\le P_{a,r}[u](x),~~~~\forall\ol B(a,r)\subset\Omega,
~~~\forall x\in B(a,r)~;$
\medskip
\item{\rm b)} $u(a)\le\mu_S\,(u\,;a,r),~~~~\forall\ol B(a,r)\subset
\Omega~;$
\medskip
\item{\rm c)} $u(a)\le\mu_B(u\,;a,r),~~~~\forall\ol B(a,r)\subset
\Omega~;$
\medskip
\item{\rm d)} for every $a\in\Omega$, there exists a sequence $(r_\nu)$
decreasing to $0$ such that
$$u(a)\le\mu_B(u\,;a,r_\nu)~~~~~\forall\nu~;$$
\medskip
\item{\rm e)} for every $a\in\Omega$, there exists a sequence $(r_\nu)$
decreasing to $0$ such that
$$u(a)\le\mu_S(u\,;a,r_\nu)~~~~~\forall\nu.$$
A function $u$ satisfying one of the above properties is said to be
subharmonic on~$\Omega$. The set of subharmonic functions will be
denoted by $\Sh(\Omega)$.
\endclaim

By (4.10) we see that a function $u\in C^2(\Omega,\bR)$ is subharmonic if
and only if $\Delta u\ge 0$~: in fact $\mu_S(u\,;\,a,r)<u(a)$ for $r$
small if $\Delta u(a)<0$. It is also clear on the definitions that every
(locally) convex function on $\Omega$ is subharmonic.

\proof{Proof} We have obvious implications
$${\rm a)}\Longrightarrow{\rm b)}\Longrightarrow{\rm c)}
\Longrightarrow{\rm d)}\Longrightarrow{\rm e)},$$ 
the second and last ones by (4.10) and the fact that
$\mu_B(u\,;a,r_\nu)\le\mu_S(u\,;a,t)$ for at least one $t\in{}]0,r_\nu[$.
In order to prove ${\rm e)}\Longrightarrow{\rm a)}$, we first
need a suitable version of the maximum principle.
\endproof

\claim{(4.13) Lemma} Let $u:\Omega\longrightarrow[-\infty,+\infty[$ be
an upper semicontinuous function satisfying property {\rm 4.12~e)}.
If $u$ attains its supremum at a point $x_0\in\Omega$, then $u$
is constant on the connected component of $x_0$ in $\Omega$.
\endclaim

\proof{Proof} We may assume that $\Omega$ is connected. Let
$$W=\{x\in\Omega~;~u(x)<u(x_0)\}.$$
$W$ is open by the upper semicontinuity, and distinct from
$\Omega$ since $x_0\notin W$. We want to show that $W=\emptyset$. 
Otherwise $W$ has a non empty connected component $W_0$, and $W_0$ has a 
boundary point $a\in\Omega$. We have $a\in\Omega\ssm W$, thus $u(a)=
u(x_0)$. By assumption 4.12$\,$e), we get $u(a)\le\mu_S(u\,;a,r_\nu)$ for
some sequence $r_\nu\to 0$. For $r_\nu$ small enough, $W_0$ intersects
$\Omega\ssm\ol B(a,r_\nu)$ and $B(a,r_\nu)$~; as $W_0$ is connected, we
also have $S(a,r_\nu)\cap W_0\ne\emptyset$. Since $u\le u(x_0)$ 
on the sphere $S(a,r_\nu)$ and $u<u(x_0)$ on its open subset 
$S(a,r_\nu)\cap W_0$, we get
$u(a)\le\mu_S(u\,;a,r)<u(x_0)$, a contradiction.\qed
\endproof

\claim{(4.14) Maximum principle} If $u$ is subharmonic in $\Omega$
$($in the sense that $u$ satisfies the weakest property {\rm 4.12$\,$e))},
then
$$\sup_\Omega u=\limsup_{\Omega\ni z\to\partial\Omega\cup\{\infty\}}u(z),$$
and $\sup_K u=\sup_{\partial K}u(z)$ for every compact subset
$K\subset\Omega$.
\endclaim

\proof{Proof} We have of course
$\limsup_{z\to\partial\Omega\cup\{\infty\}}u(z)\le\sup_\Omega u$. If
the inequality is strict, this means that the supremum is achieved on
some compact subset $L\subset\Omega$. Thus, by the upper semicontinuity,
there is $x_0\in L$ such that $\sup_\Omega u=\sup_L u=u(x_0)$.
Lemma~4.13 shows that $u$ is constant on the connected component
$\Omega_0$ of $x_0$ in~$\Omega$, hence
$$\sup_\Omega u=u(x_0)
=\limsup_{\Omega_0\ni z\to\partial\Omega_0\cup\{\infty\}}u(z)
\le\limsup_{\Omega\ni z\to\partial\Omega\cup\{\infty\}}u(z),$$
contradiction. The statement involving a compact subset $K$ is obtained by
applying the first statement to $\Omega'=K^\circ$.\qed
\endproof

\proof{Proof of $(4.12)~{\rm e)}\Longrightarrow{\rm a)}$} Let 
$u$ be an upper semicontinuous function 
satisfying 4.12~e) and $\ol B(a,r)\subset\Omega$ an arbitrary closed ball.
One can find a decreasing sequence of continuous functions
$v_k\in C^0\big(S(a,r),\bR\big)$ such that $\lim v_k=u$. Set
$h_k=P_{a,r}[v_k]\in C^0\big(\ol B(a,r),\bR\big)$. As $h_k$ is harmonic on
$B(a,r)$, the function $u-h_k$ satisfies 4.12~e) on $B(a,r)$.
Furthermore $\limsup_{x\to\xi\in S(a,r)}u(x)-h_k(x)\le u(\xi)-v_k(\xi)\le 0$,
so $u-h_k\le 0$ on $B(a,r)$ by Th.~4.14. By monotone convergence, we find
$u\le P_{a,r}[u]$ on $B(a,r)$ when $k$ tends to $+\infty$.\qed
\endproof

\subsubsection{4.C.2}{Basic Properties.} Here is a short list of the most
basic properties.

\claim{(4.15) Theorem} For any decreasing sequence $(u_k)$ of
subharmonic functions, the limit $u=\lim u_k$ is subharmonic.
\endclaim

\proof{Proof} A decreasing limit of upper semicontinuous functions is again 
upper semicontinuous, and the mean value inequalities 4.12 remain valid
for $u$ by Lebesgue's monotone convergence theorem.\qed
\endproof

\claim{(4.16) Theorem} Let $u_1,\ldots,u_p\in\Sh(\Omega)$ and
$\chi:\bR^p\longrightarrow\bR$  be a convex function such that $\chi(t_1,\ldots,
t_p)$ is non decreasing in each $t_j$. If $\chi$ is extended by continuity
into a function $[-\infty,+\infty[^p\longrightarrow[-\infty,+\infty[$, then
$$\chi(u_1,\ldots,u_p)\in\Sh(\Omega).$$
In particular~ $u_1+\cdots+u_p$, $\max\{u_1,\ldots,u_p\}$, 
$\log(e^{u_1}+\cdots+e^{u_p})\in\Sh(\Omega)$.
\endclaim

\proof{Proof} Every convex function is continuous, hence $\chi(u_1,\ldots,u_p)$
is upper semicontinuous. One can write
$$\chi(t)=\sup_{i\in I}\,A_i(t)$$
where $A_i(t)=a_1t_1+\cdots+a_pt_p+b$ is the family of affine functions that
define supporting hyperplanes of the graph of $\chi$. As $\chi(t_1,\ldots,t_p)$ 
is non-decreasing in each $t_j$, we have $a_j\ge 0$, thus
$$\sum_{1\le j\le p}a_ju_j(x)+b\le\mu_B\big(\sum a_ju_j+b\,;x,r\big)\le
\mu_B\big(\chi(u_1,\ldots,u_p)\,;x,r\big)$$
for every ball $\ol B(x,r)\subset\Omega$. If one takes the supremum of this
inequality over all the $A_i\,$'s$\,$, it follows that $\chi(u_1,\ldots,u_p)$
satisfies the mean value inequality 4.12~c). In the last example, the 
function $\chi(t_1,\ldots,t_p)=\log(e^{t_1}+\cdots+e^{t_p})$ is convex because
$$\sum_{1\le j,k\le p}{\partial^2\chi\over\partial t_j\partial t_k}\,
\xi_j\xi_k=e^{-\chi}\sum\xi_j^2\,e^{t_j}-e^{-2\chi}
\big(\sum\xi_j\,e^{t_j}\big)^2$$
and $\big(\sum\xi_j\,e^{t_j}\big)^2\le\big(\sum\xi_j^2\,e^{t_j}\big)\,e^\chi$
by the Cauchy-Schwarz inequality.\qed
\endproof

\claim{(4.17) Theorem} If $\Omega$ is connected and $u\in\Sh(\Omega)$, then
either $u\equiv-\infty$ or $u\in L^1_\loc(\Omega)$.
\endclaim

\proof{Proof} Note that a subharmonic function is always locally bounded 
above. Let $W$ be the set of points $x\in\Omega$ such that
$u$ is integrable in a neighborhood of $x$. Then $W$ is open by
definition and $u>-\infty$ almost everywhere on $W$. 
If $x\in\ol W$, one can choose $a\in W$ such that $|a-x|<r=
{1\over 2}d(x,\complement\Omega)$ and $u(a)>-\infty$. Then $B(a,r)$
is a neighborhood of $x$, $\smash{\ol B}(a,r)\subset\Omega$ and
$\mu_B(u\,;a,r)\ge u(a)>-\infty$. Therefore $x\in W$, $W$ is also closed.
We must have $W=\Omega$ or $W=\emptyset$~; in the last case $u\equiv-\infty$
by the mean value inequality.\qed
\endproof

\claim{(4.18) Theorem} Let $u\in\Sh(\Omega)$ be such that
$u\not\equiv-\infty$ on each connected component of $\Omega$. Then
\medskip
\item{\rm a)} $r\longmapsto\mu_S(u\,;a,r)$, $r\longmapsto\mu_B(u\,;a,r)$
are non decreasing functions in the interval $]0,d(a,\complement\Omega)[\,$,
and $\mu_B(u\,;a,r)\le\mu_S(u\,;a,r)$.
\smallskip
\item{\rm b)} For any family $(\rho_\varepsilon)$ of smoothing kernels,
$u\star\rho_\varepsilon\in\Sh(\Omega_\varepsilon)\cap
\cC^\infty(\Omega_\varepsilon,\bR)$, the family $(u\star\rho_\varepsilon)$
is non decreasing in $\varepsilon$ and $\lim_{\varepsilon\to 0}u\star
\rho_\varepsilon=u$.\vskip0pt
\endclaim

\proof{Proof} We first verify statements a) and b) when 
$u\in C^2(\Omega,\bR)$. Then $\Delta u\ge 0$ and $\mu_S(u\,;a,r)$
is non decreasing in virtue of (4.10). By (4.9), we find that
$\mu_B(u\,;a,r)$ is also non decreasing and that $\mu_B(u\,;a,r)\le
\mu_S(u\,;a,r)$.
Furthermore, Formula (4.11) shows that $\varepsilon\longmapsto
u\star\rho_\varepsilon(a)$ is non decreasing (provided that $\rho_\varepsilon$
is radially symmetric).

In the general case, we first observe that property 4.12~c) is
equivalent to the inequality
$$u\le u\star\mu_r~~~~\hbox{\rm on}~~\Omega_r,~~~~\forall r>0,$$
where $\mu_r$ is the probability measure of uniform density on $B(0,r)$. 
This inequality implies $u\star\rho_\varepsilon\le u\star\rho_\varepsilon\star
\mu_r$ on $(\Omega_r)_\varepsilon=\Omega_{r+\varepsilon}$, thus
$u\star\rho_\varepsilon\in \cC^\infty(\Omega_\varepsilon,\bR)$ is subharmonic on 
$\Omega_\varepsilon$.
It follows that $u\star\rho_\varepsilon\star\rho_\eta$ is non decreasing in
$\eta$~; by symmetry, it is also non decreasing in $\varepsilon$, and so is
$u\star\rho_\varepsilon=\lim_{\eta\to 0}u\star\rho_\varepsilon\star\rho_\eta$.
We have $u\star\rho_\varepsilon\ge u$ by (4.19)
and $\limsup_{\varepsilon\to 0}u\star\rho_\varepsilon\le u$ by the
upper semicontinuity. Hence $\lim_{\varepsilon\to 0}u\star\rho_\varepsilon=u$.
Property~a) for $u$ follows now from its validity for $u\star\rho_\varepsilon$
and from the monotone convergence theorem.\qed
\endproof

\claim{(4.19) Corollary} If $u\in\Sh(\Omega)$ is such that
$u\not\equiv-\infty$ on each connected component of $\Omega$, then $\Delta u$
computed in the sense of distribution theory is a positive measure.
\endclaim
 
Indeed $\Delta(u\star\rho_\varepsilon)\ge 0$ as a function, and 
$\Delta(u\star\rho_\varepsilon)$ converges
weakly to $\Delta u$ in $\cD'(\Omega)$. Corollary~4.19 has a converse,
but the correct statement is slightly more involved than for the direct
property:

\claim{(4.20) Theorem} If $v\in\cD'(\Omega)$ is such that $\Delta v$ is a
positive measure, there exists a unique function $u\in\Sh(\Omega)$ locally 
integrable such that $v$ is the distribution associated to $u$.
\endclaim

We must point out that $u$ need not coincide everywhere with $v$, even when
$v$ is a locally integrable upper semicontinuous function: for example, if
$v$ is the characteristic function of a compact subset $K\subset\Omega$
of measure $0$, the subharmonic representant of $v$ is $u=0$.

\proof{Proof} Set $v_\varepsilon=v\star\rho_\varepsilon\in
\cC^\infty(\Omega_\varepsilon,\bR)$. Then $\Delta v_\varepsilon=(\Delta v)
\star\rho_\varepsilon\ge 0$, thus $v_\varepsilon\in\Sh(\Omega_\varepsilon)$.
Arguments similar to those in the proof of Th.~4.18 show that
$(v_\varepsilon)$ is non decreasing in~$\varepsilon$. Then
$u:=\lim_{\varepsilon\to 0}~v_\varepsilon\in\Sh(\Omega)$
by Th.~4.15. Since $v_\varepsilon$ converges weakly to $v$, the
monotone convergence theorem shows that
$$\langle v,f\rangle=\lim_{\varepsilon\to 0}\int_\Omega v_\varepsilon\,f\,
d\lambda=\int_\Omega u\,f\,d\lambda,~~~~\forall f\in\cD(\Omega),~~~f\ge 0,$$
which concludes the existence part. The uniqueness of $u$ is clear from
the fact that $u$ must satisfy
$u=\lim u\star\rho_\varepsilon=\lim v\star\rho_\varepsilon$.\qed
\endproof

The most natural topology on the space $\Sh(\Omega)$ of subharmonic
functions is the topo\-logy induced by the vector space topology of
$L^1_\loc(\Omega)$ (Fr\'echet topology of convergence in $L^1$ norm on 
every compact subset of $\Omega$).

\claim{(4.21) Proposition} The convex cone $\Sh(\Omega)\cap
L^1_\loc(\Omega)$ is closed in $L^1_\loc(\Omega)$, and it has the property
that every bounded subset is relatively compact.
\endclaim

\proof{Proof} Let $(u_j)$ be a sequence in $\Sh(\Omega)\cap L^1_\loc(\Omega)$.
If $u_j\to u$ in $L^1_\loc(\Omega)$ then $\Delta u_j\to \Delta u$ in the
weak topology of distributions, hence $\Delta u\ge 0$ and $u$ can be
represented by a subharmonic function thanks to Th.~4.20. Now, suppose
that $\|u_j\|_{L^1(K)}$ is uniformly bounded for every compact subset
$K$ of~$\Omega$. Let $\mu_j=\Delta u_j\ge 0$. If $\psi\in\cD(\Omega)$
is a test function equal to $1$ on a neighborhood $\omega$ of~$K$ and
such that $0\le\psi\le 1$ on $\Omega$, we find
$$\mu_j(K)\le\int_\Omega \psi\,\Delta u_j\,d\lambda=
\int_\Omega \Delta\psi\,u_j\,d\lambda\le C\|u_j\|_{L^1(K')},$$
where $K'=\Supp\psi$, hence the sequence of measures $(\mu_j)$ is
uniformly bounded in mass on every compact subset of~$\Omega$. By weak
compactness, there is a subsequence $(\mu_{j_\nu})$ which converges
weakly to a positive measure $\mu$ on~$\Omega$. We claim that $f\star
(\psi\mu_{j_\nu})$ converges to $f\star(\psi\mu)$ in $L^1_\loc(\bR^m)$
for every function $f\in L^1_\loc(\bR^m)$. In fact, this is clear if
$f\in \cC^\infty(\bR^m)$, and in general we use an approximation of
$f$ by a smooth function $g$ together with the estimate
$$\|(f-g)\star(\psi\mu_{j_\nu})\|_{L^1(A)}\le\|(f-g)\|_{L^1(A+K')}
\mu_{j_\nu}(K'),\qquad\forall A\compact\bR^m$$
to get the conclusion. We apply this when $f=N$ is the Newton kernel. Then
$h_j=u_j-N\star(\psi\mu_j)$ is harmonic on $\omega$ and bounded in
$L^1(\omega)$. As $h_j=h_j\star\rho_\varepsilon$ for any smoothing kernel
$\rho_\varepsilon$, we see that all derivatives $D^\alpha h_j=h_j\star
(D^\alpha\rho_\varepsilon)$ are in fact uniformly locally bounded
in~$\omega$. Hence, after extracting a new subsequence, we may suppose
that $h_{j_\nu}$ converges uniformly to a limit $h$ on~$\omega$. Then
$u_{j_\nu}=h_{j_\nu}+N\star(\psi\mu_{j_\nu})$ converges to
$u=h+N\star(\psi\mu)$ in $L^1_\loc(\omega)$, as desired.\qed
\endproof

We conclude this subsection by stating a generalized version of the
Green-Riesz formula.

\claim{(4.22) Proposition} Let $u\in\Sh(\Omega)\cap L^1_\loc(\Omega)$
and $\ol B(0,r)\subset\Omega$.
\medskip
\item{\rm a)} The Green-Riesz formula still holds true for such an $u$,
namely, for every $x\in B(0,r)$
$$u(x)=\int_{B(0,r)}\Delta u(y)\,G_r(x,y)\,d\lambda(y)+\int_{S(0,r)}
u(y)\,P_r(x,y)\,d\sigma(y).$$
\item{\rm b)} {\rm (Harnack inequality)}\Fbreak
If $u\ge 0$ on $\ol B(0,r)$, then for all $x\in B(0,r)$
$$0\le u(x)\le\int_{S(0,r)}u(y)\,P_r(x,y)\,d\sigma(y)\le
{r^{m-2}(r+|x|)\over(r-|x|)^{m-1}}\,\mu_S(u\,;0,r).$$
If $u\le 0$ on $\ol B(0,r)$, then for all $x\in B(0,r)$
$$u(x)\le\int_{S(0,r)}u(y)\,P_r(x,y)\,d\sigma(y)\le
{r^{m-2}(r-|x|)\over(r+|x|)^{m-1}}\,\mu_S(u\,;0,r)\le 0.$$
\vskip0pt
\endclaim

\proof{Proof} We know that a) holds true if $u$ is of class $C^2$.
In general, we replace $u$ by $u\star\rho_\varepsilon$ and
take the limit. We only have to check that
$$\int_{B(0,r)}\mu\star\rho_\varepsilon(y)\,G_r(x,y)\,d\lambda(y)=
\lim_{\varepsilon\to 0}\int_{B(0,r)}\mu(y)\,G_r(x,y)\,d\lambda(y)$$
for the positive measure $\mu=\Delta u$. Let us denote by $\wt G_x(y)$
the function such that
$$\wt G_x(y)=\cases{
G_r(x,y)&if $x\in B(0,r)$\cr
0&if $x\notin B(0,r)$.\cr}$$
Then
$$\eqalign{
\int_{B(0,r)}\mu\star\rho_\varepsilon(y)\,G_r(x,y)\,d\lambda(y)
&=\int_{\bR^m}\mu\star\rho_\varepsilon(y)\,\wt G_x(y)\,d\lambda(y)\cr
&=\int_{\bR^m}\mu(y)\,\wt G_x\star\rho_\varepsilon(y)\,d\lambda(y).\cr}$$
However $\wt G_x$ is continuous on $\bR^m\ssm\{x\}$ and subharmonic
in a neighborhood of~$x$, hence $\wt G_x\star\rho_\varepsilon$ converges
uniformly to $\wt G_x$ on every compact subset of $\bR^m\ssm\{x\}$, and
converges pointwise monotonically in a neighborhood of~$x$. The desired
equality follows by the monotone convergence theorem. Finally, b) is a
consequence of a), for the integral involving $\Delta u$ is nonpositive and
$${1\over\sigma_{m-1}r^{m-1}}{r^{m-2}(r-|x|)\over (r+|x|)^{m-1}}\le
P_r(x,y)\le {1\over\sigma_{m-1}r^{m-1}}{r^{m-2}(r+|x|)\over(r-|x|)^{m-1}}$$
by (4.6) combined with the obvious inequality
$(r-|x|)^m\le |x-y|^m\le (r+|x|)^m$.\qed
\endproof

\subsubsection{4.C.3}{Upper Envelopes and Choquet's Lemma.} 
Let $\Omega\subset\bR^n$ and let $(u_\alpha)_{\alpha\in I}$ be a family 
of upper semicontinuous functions $\Omega\longrightarrow[-\infty,+\infty[$. 
We assume that $(u_\alpha)$ is locally uniformly bounded above. 
Then the upper envelope
$$u=\sup u_\alpha$$
need not be upper semicontinuous, so we consider its {\it upper
semicontinuous regularization}\/:
$$u^\star(z)=\lim_{\varepsilon\to 0}\sup_{B(z,\varepsilon)}u\ge u(z).$$
It is easy to check that $u^\star$ is the smallest upper semicontinuous
function which is${}\ge u$. Our goal is to show that $u^\star$ can be
computed with a countable subfamily of $(u_\alpha)$. Let
$B(z_j,\varepsilon_j)$ be a countable basis of the topology
of~$\Omega$. For each $j$, let $(z_{jk})$ be a sequence of points
in $B(z_j,\varepsilon_j)$ such that
$$\sup_k u(z_{jk})=\sup_{B(z_j,\varepsilon_j)}u,$$
and for each pair $(j,k)$, let $\alpha(j,k,l)$ be a sequence of indices
$\alpha\in I$ such that $u(z_{jk})=\sup_l u_{\alpha(j,k,l)}(z_{jk})$. Set
$$v=\sup_{j,k,l}u_{\alpha(j,k,l)}.$$
Then $v\le u$ and $v^\star\le u^\star$. On the other hand
$$\sup_{B(z_j,\varepsilon_j)}v\ge\sup_k v(z_{jk})\ge\sup_{k,l}
u_{\alpha(j,k,l)}(z_{jk})=\sup_k u(z_{jk})=\sup_{B(z_j,\varepsilon_j)}u.$$
As every ball $B(z,\varepsilon)$ is a union of balls
$B(z_j,\varepsilon_j)$, we easily conclude that $v^\star\ge u^\star$,
hence $v^\star=u^\star$. Therefore:

\claim{(4.23) Choquet's lemma} Every family $(u_\alpha)$ has a
countable subfamily $(v_j)=(u_{\alpha(j)})$ such that its upper envelope
$v$ satisfies $v\le u\le u^\star=v^\star$.\qed
\endclaim

\claim{(4.24) Proposition} If all $u_\alpha$ are subharmonic,
the upper regularization $u^\star$ is
subharmonic and equal almost everywhere to $u$.
\endclaim

\proof{Proof} By Choquet's lemma we may assume that $(u_\alpha)$ is countable.
Then $u=\sup u_\alpha$ is a Borel function.
As each $u_\alpha$ satisfies the mean value inequality on every
ball $\ol B(z,r)\subset\Omega$, we get
$$u(z)=\sup u_\alpha(z)\le\sup\mu_B(u_\alpha\,;\,z,r)\le\mu_B(u\,;\,z,r).$$
The right-hand side is a continuous function of $z$, so we infer
$$u^\star(z)\le\mu_B(u\,;\,z,r)\le\mu_B(u^\star\,;\,z,r)$$
and $u^\star$ is subharmonic. By the upper semicontinuity of $u^\star$
and the above inequality we find $u^\star(z)=\lim_{r\to 0}\mu_B(u\,;\,z,r)$,
thus $u^\star=u$ almost everywhere by Lebesgue's lemma.\qed
\endproof

\section{5}{Plurisubharmonic Functions}
\subsection{5.A}{Definition and Basic Properties}
Plurisubharmonic functions have been introduced independently
by \cite{Lelong 1942} and \cite{Oka 1942} for the study of 
holomorphic convexity.
They are the complex counterparts of subharmonic functions.

\claim{(5.1) Definition} A function $u:\Omega\longrightarrow[-\infty,+\infty[$ defined
on an open subset $\Omega\subset\bC^n$ is said to be plurisubharmonic if
\medskip
\item{\rm a)} $u$ is upper semicontinuous~$;$
\smallskip
\item{\rm b)} for every complex line $L\subset\bC^n$, 
$u_{\restriction\Omega\cap L}$ is subharmonic on $\Omega\cap L$.
\smallskip
\noindent The set of plurisubharmonic functions on $\Omega$ is denoted by
$\Psh(\Omega)$.
\vskip0pt
\endclaim

An equivalent way of stating property b) is: for all
$a\in\Omega$, $\xi\in\bC^n$, $|\xi|<d(a,\complement\Omega)$, then
$$u(a)\le{1\over 2\pi}\int_0^{2\pi}u(a+e^{\ii\theta}\,\xi)\,d\theta.
\leqno(5.2)$$
An integration of (5.2) over $\xi\in S(0,r)$ yields $u(a)\le\mu_S(u\,;a,r)$,
therefore 
$$\Psh(\Omega)\subset\Sh(\Omega).\leqno(5.3)$$
The following results have already been proved for subharmonic functions
and are easy to extend to the case of plurisubharmonic functions:

\claim{(5.4) Theorem} For any decreasing sequence of plurisubharmonic
functions $u_k\in\Psh(\Omega)$, the limit $u=\lim u_k$ is plurisubharmonic
on $\Omega$.
\endclaim

\claim{(5.5) Theorem} Let $u\in\Psh(\Omega)$ be such that $u\not\equiv-\infty$ 
on every connected component of $\Omega$. If $(\rho_\varepsilon)$ is a 
family of smoothing kernels, then $u\star\rho_\varepsilon$ is $\cC^\infty$ and 
plurisubharmonic on $\Omega_\varepsilon$, the family 
$(u\star\rho_\varepsilon)$ is non decreasing in $\varepsilon$ and 
$\lim_{\varepsilon\to 0}u\star\rho_\varepsilon=u$.
\endclaim

\claim{(5.6) Theorem} Let $u_1,\ldots,u_p\in\Psh(\Omega)$ and $\chi:\bR^p\longrightarrow\bR$
be a convex function such that $\chi(t_1,\ldots,t_p)$ is non decreasing in each
$t_j$. Then $\chi(u_1,\ldots,u_p)$ is plurisubharmonic on $\Omega$. In 
particular~ $u_1+\cdots+u_p$, $\max\{u_1,\ldots,u_p\}$, 
$\log(e^{u_1}+\cdots+e^{u_p})$ are plurisubharmonic on $\Omega$.
\endclaim

\claim{(5.7) Theorem} Let $\{u_\alpha\}\subset\Psh(\Omega)$ be locally
uniformly bounded from above and $u=\sup u_\alpha$. Then the 
regularized upper envelope $u^\star$ is plurisubharmonic
and is equal to $u$ almost everywhere.
\endclaim

\proof{Proof} By Choquet's lemma, we may assume that $(u_\alpha)$ is countable.
Then $u$ is a Borel function which clearly satisfies (5.2), and
thus $u\star\rho_\varepsilon$ also satisfies (5.2). Hence
$u\star\rho_\varepsilon$ is plurisubharmonic. By Proposition~4.24,
$u^\star=u$ almost everywhere and $u^\star$ is subharmonic, so
$$u^\star=\lim u^\star\star\rho_\varepsilon=\lim u\star\rho_\varepsilon$$
is plurisubharmonic.\qed
\endproof

If $u\in C^2(\Omega,\bR)$, the subharmonicity of restrictions of $u$ to
complex lines, $\bC\ni w\longmapsto u(a+w\xi)$, $a\in\Omega$,
$\xi\in\bC^n$, is equivalent to
$${\partial^2\over\partial w\partial\ol w}u(a+w\xi)=\sum_{1\le j,k\le n}\,
{\partial^2 u\over\partial z_j\partial\ol z_k}(a+w\xi)\,\xi_j\ol\xi_k\ge 0.$$
Therefore, $u$ is plurisubharmonic on $\Omega$ if and only if 
$\sum\partial^2 u/\partial z_j\partial\ol z_k(a)\,\xi_j\ol\xi_k$
is a semiposi\-tive hermitian form
at every point $a\in\Omega$. This equivalence is still true
for arbitrary plurisubharmonic functions, under the following form:

\claim{(5.8) Theorem} If $u\in\Psh(\Omega)$, $u\not\equiv-\infty$ on every
connected component of $\Omega$, then for all $\xi\in\bC^n$
$$Hu(\xi):=\sum_{1\le j,k\le n}{\partial^2 u\over\partial z_j\partial\ol z_k}
\,\xi_j\ol\xi_k\in\cD'(\Omega)$$
is a positive measure. Conversely, if $v\in\cD'(\Omega)$ is such that 
$Hv(\xi)$ is a positive measure for every $\xi\in\bC^n$, there exists a 
unique function $u\in\Psh(\Omega)$ locally integrable on $\Omega$ such that 
$v$ is the distribution associated to $u$.
\endclaim

\proof{Proof} If $u\in\Psh(\Omega)$, then
$Hu(\xi)={\rm weak}~\lim\,H(u\star\rho_\varepsilon)(\xi)\ge 0$. Conversely,
$Hv\ge 0$ implies $H(v\star\rho_\varepsilon)=(Hv)\star\rho_\varepsilon\ge 0$,
thus $v\star\rho_\varepsilon\in\Psh(\Omega)$, and also $\Delta v\ge 0$,
hence $(v\star\rho_\varepsilon)$ is non decreasing in $\varepsilon$ and
$u=\lim_{\varepsilon\to 0}v\star\rho_\varepsilon\in\Psh(\Omega)$
by Th.~5.4.\qed
\endproof

\claim{(5.9) Proposition} The convex cone $\Psh(\Omega)\cap
L^1_\loc(\Omega)$ is closed in $L^1_\loc(\Omega)$, and it has the property
that every bounded subset is relatively compact.
\endclaim

\subsection{5.B}{Relations with Holomorphic Functions}
In order to get a better geometric insight, we assume more generally that $u$
is a $C^2$ function on a complex $n$-dimensional manifold~$X$.
The {\it complex Hessian} of $u$ at a point $a\in X$ is the hermitian form
on $T_X$ defined by
$$Hu_a=\sum_{1\le j,k\le n}\,{\partial^2 u\over\partial z_j\partial\ol z_k}(a)
\,dz_j\otimes d\ol z_k.\leqno(5.10)$$
If $F:X\longrightarrow Y$ is a holomorphic mapping and if $v\in C^2(Y,\bR)$, we have
$d'd''(v\circ F)=F^\star d'd''v$. In equivalent notations, a direct
calculation gives for all $\xi\in T_{X,a}$
$$H(v\circ F)_a(\xi)=\sum_{j,k,l,m}{\partial^2 v\big(F(a)\big)\over
\partial z_l\partial\ol z_m}\,{\partial F_l\big(a)\over\partial z_j}\xi_j\,
\ol{{\partial F_m\big(a)\over\partial z_k}\xi_k}
=Hv_{F(a)}\big(F'(a).\xi\big).$$
In particular $Hu_a$ does not depend on the choice of coordinates
$(z_1,\ldots,z_n)$ on $X$, and $Hv_a\ge 0$ on $Y$ implies $H(v\circ F)_a\ge 0$
on $X$. Therefore, the notion of plurisubharmonic function makes sense 
on any complex manifold.

\claim{(5.11) Theorem} If $F:X\longrightarrow Y$ is a holomorphic map and
$v\in\Psh(Y)$, then $v\circ F\in\Psh(X)$.
\endclaim

\proof{Proof} It is enough to prove the result when $X=\Omega_1\subset\bC^n$
and $X=\Omega_2\subset\bC^p$ are open subsets . The conclusion is already known 
when $v$ is of class $C^2$, and it can be extended to an arbitrary upper
semicontinuous function $v$ by using Th.~5.4 and the fact that 
$v=\lim v\star\rho_\varepsilon$.\qed
\endproof

\claim{(5.12) Example} \rm By (3.22) we see that $\log|z|$ is subharmonic
on $\bC$, thus $\log|f|\in\Psh(X)$ for every holomorphic function 
$f\in\cO(X)$. More generally 
$$\log\big(|f_1|^{\alpha_1}+\cdots+|f_q|^{\alpha_q}\big)\in\Psh(X)$$
for every $f_j\in\cO(X)$ and $\alpha_j\ge 0$ (apply Th.~5.6 with
$u_j=\alpha_j\,\log|f_j|~$).
\endclaim

\subsection{5.C}{Convexity Properties}
The close analogy of plurisubharmonicity with the concept of convexity
strongly suggests that there are deeper connections between these notions.
We describe here a few elementary facts illustrating this philosophy.
Another interesting connection between plurisubharmonicity and convexity
will be seen in \S~7.B (Kiselman's minimum principle).

\claim{(5.13) Theorem} If $\Omega=\omega+\ii\omega'$ where $\omega$,
$\omega'$ are open subsets of $\bR^n$, and if $u(z)$ is a
plurisubharmonic function on $\Omega$ that depends only on $x=\Re z$,
then $\omega\ni x\longmapsto u(x)$ is convex.
\endclaim

\proof{Proof} This is clear when $u\in C^2(\Omega,\bR)$, for
$\partial^2u/\partial z_j\partial\ol z_k={1\over 4}\,\partial^2u/
\partial x_j\partial x_k$. In the general case, write $u=\lim u\star
\rho_\varepsilon$ and observe that $u\star\rho_\varepsilon(z)$ depends only
on $x$.\qed
\endproof

\claim{(5.14) Corollary} If $u$ is a plurisubharmonic function in the open
polydisk $D(a,R)=\prod D(a_j,R_j)\subset\bC^n$, then
$$\eqalign{
\mu(u\,;\,r_1,\ldots,r_n)&={1\over(2\pi)^n}\int_0^{2\pi}u(a_1+r_1e^{\ii\theta_1}
,\ldots,a_n+r_ne^{\ii\theta_n})\,d\theta_1\ldots d\theta_n,\cr
m(u\,;\,r_1,\ldots,r_n)&=\sup_{z\in D(a,r)}u(z_1,\ldots,z_n),~~~~r_j<R_j\cr}$$
are convex functions of $(\log r_1,\ldots,\log r_n)$ that are non decreasing in
each variable.
\endclaim

\proof{Proof} That $\mu$ is non decreasing follows from the subharmonicity of
$u$ along every coordinate axis. Now, it is easy to verify that the functions
$$\eqalign{
\wt\mu(z_1,\ldots,z_n)&={1\over(2\pi)^n}\int_0^{2\pi}u(a_1+e^{z_1}
e^{\ii\theta_1},\ldots,a_n+e^{z_n}e^{\ii\theta_n})\,d\theta_1\ldots d\theta_n,\cr
\wt m(z_1,\ldots,z_n)&=\sup_{|w_j|\le1}u(a_1+e^{z_1}w_1,\ldots,a_n+e^{z_n}w_n)\cr}
$$
are upper semicontinuous, satisfy the mean value inequality, and depend only 
on $\Re z_j\in{}]0,\log R_j[$. Therefore $\wt\mu$ and $\wt M$ are convex.
Cor.~5.14 follows from the equalities
$$\eqalignno{
\mu(u\,;\,r_1,\ldots,r_n)&=\wt\mu(\log r_1,\ldots,\log r_n),\cr
m(u\,;\,r_1,\ldots,r_n)&=\wt m(\log r_1,\ldots,\log r_n).&\square\cr}$$
\endproof

\subsection{5.D}{Pluriharmonic Functions}
Pluriharmonic functions are the counterpart of harmonic functions in the
case of functions of complex variables:

\claim{(5.15) Definition} A function $u$ is said to be pluriharmonic if 
$u$ and $-u$ are plurisubharmonic.
\endclaim

A pluriharmonic function is harmonic (in particular smooth) in any
$\bC$-analytic coordinate system, and is characterized by the condition 
$Hu=0$, i.e.\ $d'd''u=0$ or
$$\partial^2u/\partial z_j\partial\ol z_k=0~~~\hbox{\rm for~all~~}j,k.$$
If $f\in\cO(X)$, it follows that the functions $\Re f,~\Im f$ are 
pluriharmonic. Conversely:

\claim{(5.16) Theorem} If the De Rham cohomology group $H^1_{\DR}(X,\bR)$
is zero, every pluriharmonic function
$u$ on $X$ can be written $u=\Re f$ where $f$ is a holomorphic
function on $X$.
\endclaim

\proof{Proof} By hypothesis $H^1_{\DR}(X,\bR)=0$, $u\in \cC^\infty(X)$ and
$d(d'u)=d''d'u=0$, hence there exists $g\in \cC^\infty(X)$ such that $dg=d'u$.
Then $dg$ is of type $(1,0)$, i.e.\ $g\in\cO(X)$ and
$$d(u-2\Re g)=d(u-g-\ol g)=(d'u-dg)+(d''u-d\ol g)=0.$$
Therefore $u=\Re(2g+C)$, where $C$ is a locally constant function.\qed
\endproof

\subsection{5.E}{Global Regularization of Plurisubharmonic Functions} 
We now study a very efficient regularization and patching procedure
for continuous plurisubharmonic functions, essentially due to
\cite{Richberg 1968}. The main idea is contained in the following lemma:

\claim{(5.17) Lemma} Let $u_\alpha\in\Psh(\Omega_\alpha)$ where
$\Omega_\alpha\compact X$ is a locally finite open covering of~$X$.
Assume that for every index $\beta$
$$\limsup_{\zeta\to z}u_\beta(\zeta)<\max_{\Omega_\alpha\ni z}
\{u_\alpha(z)\}$$
at all points $z\in\partial\Omega_\beta$. Then the function
$$u(z)=\max_{\Omega_\alpha\ni z}~u_\alpha(z)$$
is plurisubharmonic on $X$.
\endclaim

\proof{Proof} Fix $z_0\in X$. Then the indices $\beta$ such that $z_0\in
\partial\Omega_\beta$ or $z_0\notin\ol\Omega_\beta$ do not contribute 
to the maximum in a neighborhood of $z_0$. Hence there is a
a finite set $I$ of indices $\alpha$ such that
$\Omega_\alpha\ni z_0$ and a neighborhood $V\subset\bigcap_{\alpha\in I}
\Omega_\alpha$ on which $u(z)=\max_{\alpha\in I}u_\alpha(z)$.
Therefore $u$ is plurisubharmonic on $V$.\qed
\endproof

The above patching procedure produces functions which are in general
only continuous. When smooth functions are needed, one has to use a
regularized max function. Let $\theta\in \cC^\infty(\bR,\bR)$ be a
nonnegative function with support in $[-1,1]$ such that
$\int_\bR\theta(h)\,dh=1$ and $\int_\bR h\theta(h)\,dh=0$.

\claim{(5.18) Lemma} For arbitrary $\eta=(\eta_1,\ldots,\eta_p)\in{}
]0,+\infty[^p$, the function
$$M_{\eta}(t_1,\ldots,t_p)=\int_{\bR^n}\max\{t_1+h_1,\ldots,t_p+h_p\}
\prod_{1\le j\le n}\theta(h_j/\eta_j)\,dh_1\ldots dh_p$$
possesses the following properties:
\medskip
\item{\rm a)} $M_\eta(t_1,\ldots,t_p)$ is non decreasing in 
all variables, smooth and convex on $\bR^n~;$
\smallskip
\item{\rm b)} $\max\{t_1,\ldots,t_p\}\le M_\eta(t_1,\ldots,t_p)\le
\max\{t_1+\eta_1,\ldots,t_p+\eta_p\}~;$
\smallskip
\item{\rm c)} $\,M_\eta(t_1,\ldots,t_p)=M_{(\eta_1,\ldots,\wh{\eta_j},\ldots,\eta_p)}
(t_1,\ldots,\wh{t_j},,\ldots,t_p)$\\
if $t_j+\eta_j\le\max_{k\ne j}\{t_k-\eta_k\}~;$
\smallskip
\item{\rm d)} $M_\eta(t_1+a,\ldots,t_p+a)=M_\eta(t_1,\ldots,t_p)+a$,~~~~
$\forall a\in\bR~;$
\smallskip
\item{\rm e)} if $u_1,\ldots,u_p$ are plurisubharmonic and satisfy
$H(u_j)_z(\xi)\ge\gamma_z(\xi)$ where $z\mapsto\gamma_z$ is a continuous
hermitian form on $T_X$, then $u=M_\eta(u_1,\ldots,u_p)$ is plurisubharmonic
and satisfies $Hu_z(\xi)\ge\gamma_z(\xi)$.
\vskip0pt
\endclaim

\proof{Proof} The change of variables $h_j\mapsto h_j-t_j$ shows that $M_\eta$
is smooth. All properties are immediate consequences of the definition,
except perhaps e). That $M_\eta(u_1,\ldots,u_p)$ is plurisubharmonic
follows from a) and Th.~5.6. Fix a point $z_0$ and $\varepsilon>0$.
All functions $u'_j(z)=u_j(z)-\gamma_{z_0}(z-z_0)+\varepsilon|z-z_0|^2$
are plurisubharmonic near $z_0$. It follows that
$$M_\eta(u'_1,\ldots,u'_p)=u-\gamma_{z_0}(z-z_0)+\varepsilon|z-z_0|^2$$
is also plurisubharmonic near $z_0$. Since $\varepsilon>0$ was
arbitrary, e) follows.\qed
\endproof

\claim{(5.19) Corollary} Let $u_\alpha\in \cC^\infty(\ol\Omega_\alpha)\cap
\Psh(\Omega_\alpha)$ where $\Omega_\alpha\compact X$ is a locally finite
open covering of $X$. Assume that $u_\beta(z)<\max\{u_\alpha(z)\}$ 
at every point $z\in\partial\Omega_\beta$,
when $\alpha$ runs over the indices such that $\Omega_\alpha\ni z$.
Choose a family $(\eta_\alpha)$ of positive numbers so small that 
$u_\beta(z)+\eta_\beta\le\max_{\Omega_\alpha\ni z}
\{u_\alpha(z)-\eta_\alpha\}$ for all $\beta$ and $z\in\partial\Omega_\beta$.
Then the function defined by
$$\wt u(z)=M_{(\eta_\alpha)}\big(u_\alpha(z)\big)~~~~
{\rm for~\alpha~such~that}~~\Omega_\alpha\ni z$$
is smooth and plurisubharmonic on $X$.\qed
\endclaim

\claim{(5.20) Definition} A function $u\in\Psh(X)$ is said to be strictly
plurisubharmonic if $u\in L^1_\loc(X)$ and if for every point
$x_0\in X$ there exists a neighborhood $\Omega$ of $x_0$ and $c>0$ such 
that $u(z)-c|z|^2$ is plurisubharmonic on $\Omega$, i.e.\
$\sum~({\partial^2u/\partial z_j\partial\ol z_k})\xi_j\ol\xi_k\ge
c|\xi|^2$ $($as distributions on $\Omega)$ for all $\xi\in\bC^n$.
\endclaim

\claim{(5.21) Theorem {\rm(\cite{Richberg 1968})}} Let $u\in\Psh(X)$ be a
continuous function which is strictly plurisubharmonic on an open subset
$\Omega\subset X$, with $Hu\ge\gamma$ for some continuous positive
hermitian form $\gamma$ on $\Omega$.  For any continuous function
$\lambda\in C^0(\Omega)$, $\lambda>0$, there exists a plurisubharmonic
function $\wt u$ in $C^0(X)\cap \cC^\infty(\Omega)$ such that $u\le\wt u\le
u+\lambda$ on $\Omega$ and $\wt u=u$ on $X\ssm\Omega$, which is
strictly plurisubharmonic on $\Omega$ and satisfies $H\wt
u\ge(1-\lambda)\gamma$. In particular, $\wt u$ can be chosen
strictly plurisubharmonic on $X$ if $u$ has the same property.
\endclaim

\proof{Proof} Let $(\Omega_\alpha)$ be a locally finite
open covering of $\Omega$ by relatively compact open balls contained
in coordinate patches of $X$. Choose concentric balls
$\Omega''_\alpha\subset\Omega'_\alpha\subset\Omega_\alpha$ of respective
radii $r''_\alpha<r'_\alpha<r_\alpha$ and center $z=0$ in the given 
coordinates $z=(z_1,\ldots,z_n)$ near $\ol\Omega_\alpha$,
such that $\Omega''_\alpha$ still cover $\Omega$. We set
$$u_\alpha(z)=u\star\rho_{\varepsilon_\alpha}(z)+
\delta_\alpha(r^{\prime 2}_\alpha-|z|^2)~~~~\hbox{\rm on}~~\ol\Omega_\alpha.$$
For $\varepsilon_\alpha<\varepsilon_{\alpha,0}$ and 
$\delta_\alpha<\delta_{\alpha,0}$ small enough, we have
$u_\alpha\le u+\lambda/2$ and $Hu_\alpha\ge(1-\lambda)\gamma$ on
$\ol\Omega_\alpha$.  Set
$$\eta_\alpha=\delta_\alpha\,\min\{r^{\prime 2}_\alpha-r^{\prime\prime
2}_\alpha,(r^2_\alpha-r^{\prime2}_\alpha)/2\}.$$ 
Choose first $\delta_\alpha<\delta_{\alpha,0}$ such that $\eta_\alpha<
\min_{\ol\Omega_\alpha}\lambda/2$, and then
$\varepsilon_\alpha<\varepsilon_{\alpha,0}$ so small that
$u\le u\star\rho_{\varepsilon_\alpha}<u+\eta_\alpha$ on $\ol\Omega_\alpha$. 
As $\delta_\alpha(r^{\prime 2}-|z|^2)$ is $\le-2\eta_\alpha$ on
$\partial\Omega_\alpha$ and $>\eta_\alpha$ on $\ol\Omega''_\alpha$,
we have $u_\alpha<u-\eta_\alpha$ on $\partial\Omega_\alpha$ and
$u_\alpha>u+\eta_\alpha$ on $\ol\Omega''_\alpha$, so that the condition
required in Corollary 5.19 is satisfied. We define
$$\wt u=\cases{u&on~~$X\ssm\Omega$,\cr
               M_{(\eta_\alpha)}(u_\alpha)&on~~$\Omega$.\cr}$$
By construction, $\wt u$ is smooth on $\Omega$ and satisfies
$u\le\wt u\le u+\lambda$, $Hu\ge(1-\lambda)\gamma$ thanks to
5.18~(b,e). In order to see that $\wt u$ is plurisubharmonic on $X$,
observe that $\wt u$ is the uniform limit of $\wt u_\alpha$ with
$$\wt u_\alpha=\max\big\{u\,,\,M_{(\eta_1\ldots\eta_\alpha)}
(u_1\ldots u_\alpha)\big\}~~~\hbox{\rm on}~~\bigcup_{1\le\beta\le\alpha}
\Omega_\beta$$
and $\wt u_\alpha=u$ on the complement.\qed
\endproof

\subsection{5.F}{Polar and Pluripolar Sets.}
Polar and pluripolar sets are sets of $-\infty$ poles of subharmonic and
plurisubharmonic functions. Although these functions possess a large
amount of flexi\-bility, pluripolar sets have some properties which remind 
their loose relationship with holomorphic functions.

\claim{(5.22) Definition} A set $A\subset\Omega\subset\bR^m$
$($resp.\ $A\subset X,$ ${\rm dim}_\bC X=n)$ is said to be polar
$($resp.\ pluripolar$)$ if for every point $x\in\Omega$ there exist a
connected neighborhood $W$ of $x$ and $u\in\Sh(W)$ $($resp.\ $u\in\Psh(W))$,
$u\not\equiv-\infty$, such that~
$A\cap W\subset\{x\in W~;~u(x)=-\infty\}$.
\endclaim

Theorem~4.17 implies that a polar or pluripolar set is of zero Lebesgue 
measure. Now, we prove a simple extension theorem.

\claim{(5.23) Theorem} Let $A\subset\Omega$ be a closed polar set and
$v\in\Sh(\Omega\ssm A)$ such that $v$ is bounded above in a
neighborhood of every point of $A$. Then $v$ has a unique extension 
$\wt v\in\Sh(\Omega)$.
\endclaim

\proof{Proof} The uniqueness is clear because $A$ has zero Lebesgue measure.
On the other hand, every point of $A$ has a neighborhood $W$ such that
$$A\cap W\subset\{x\in W~;~u(x)=-\infty\},~~~~u\in\Sh(W),~~~
u\not\equiv-\infty.$$
After shrinking $W$ and subtracting a constant to $u$, we may assume
$u\le 0$.  Then for every $\varepsilon>0$ the function
$v_\varepsilon=v+\varepsilon u \in\Sh(W\ssm A)$ can be extended as
an upper semicontinuous on $W$ by setting $v_\varepsilon=-\infty$ on
$A\cap W$.  Moreover, $v_\varepsilon$ satisfies the mean value
inequality $v_\varepsilon(a)\le\mu_S (v_\varepsilon\,;a,r)$ if $a\in
W\ssm A$, $r<d(a,A\cup\complement W)$, and also clearly if $a\in
A$, $r<d(a,\complement W)$.  Therefore $v_\varepsilon\in\Sh(W)$ and
$\wt v=(\sup v_\varepsilon)^\star\in\Sh(W)$. Clearly $\wt v$
coincides with $v$ on $W\ssm A$. A similar proof gives:
\endproof

\claim{(5.24) Theorem} Let $A$ be a closed pluripolar set in a complex
analytic manifold $X$. Then every function $v\in\Psh(X\ssm A)$ that
is locally bounded above near $A$ extends uniquely into a function
$\wt v\in\Psh(X)$.\qed
\endclaim

\claim{(5.25) Corollary} Let $A\subset X$ be a closed pluripolar set. Every
holomorphic function $f\in\cO(X\ssm A)$ that is locally bounded near
$A$ extends to a holomorphic function $\wt f\in\cO(X)$.
\endclaim

\proof{Proof} Apply Th.~5.24 to $\pm\Re f$ and $\pm\Im f$.
It follows that $\Re f$ and $\Im f$ have pluriharmonic extensions to $X$,
in particular $f$ extends to $\wt f\in \cC^\infty(X)$. By density of
$X\ssm A$, $d''\wt f=0$ on $X$.\qed
\endproof

\claim{(5.26) Corollary} Let $A\subset\Omega$ $($resp.\ $A\subset X)$ be a closed
$($pluri$)$polar set. If $\Omega$ $($resp.\ $X)$ is connected, then
$\Omega\ssm A$ $($resp.\ $X\ssm A)$ is connected.
\endclaim

\proof{Proof} If $\Omega\ssm A$ $($resp.\ $X\ssm A$) is a disjoint
union $\Omega_1\cup\Omega_2$ of non empty open subsets, the function defined
by $f\equiv 0$ on $\Omega_1$, $f\equiv 1$ on $\Omega_2$ would have a
harmonic (resp.\ holomorphic) extension through $A$, a contradiction.\qed
\endproof

\section{6}{Domains of Holomorphy and Stein Manifolds}
\subsection{6.A}{Domains of Holomorphy in $\bC^n$. Examples}
Loosely speaking, a domain of holomorphy is an open subset $\Omega$ in
$\bC^n$ such that there is no part of $\partial\Omega$ across which
all functions  $f\in\cO(\Omega)$ can be extended. More precisely:

\claim{(6.1) Definition} Let $\Omega\subset\bC^n$ be an open
subset. $\Omega$ is said to be a domain of holomorphy if for every
connected open set  $U\subset\bC^n$ which meets $\partial\Omega$ and
every connected component $V$  of $U\cap\Omega$ there exists
$f\in\cO(\Omega)$ such that $f_{\restriction V}$ has no holomorphic
extension to $U$.
\endclaim

Under the hypotheses made on $U$, we have $\emptyset\ne\partial V\cap
U\subset\partial\Omega$. In order to show that $\Omega$ is a domain
of holomorphy, it is thus sufficient to find for every $z_0\in
\partial\Omega$ a function $f\in\cO(\Omega)$ which is unbounded near~$z_0$.

\claim{(6.2) Examples} \rm Every open subset $\Omega\subset\bC$ is a domain
of holomorphy (for any $z_0\in\partial\Omega$, $f(z)=(z-z_0)^{-1}$ cannot be
extended at $z_0\,$). In $\bC^n$, every {\it convex} open subset is a domain
of holomorphy: if $\Re\langle z-z_0,\xi_0\rangle=0$ is a supporting hyperplane 
of $\partial\Omega$ at $z_0$, the function $f(z)=(\langle z-z_0,\xi_0
\rangle)^{-1}$ is holomorphic on $\Omega$ but cannot be extended at $z_0$.
\endclaim

\claim{(6.3) Hartogs figure} \rm
Assume that $n\ge 2$. Let $\omega\subset\bC^{n-1}$ be a connected open 
set and $\omega'\subsetneq\omega$ an
open subset. Consider the open sets in $\bC^n$~:
$$\cmalign{
&\Omega=\big((D(R)\ssm\ol D(r))\times\omega\big)\cup
\big(D(R)\times\omega'\big)~~~~&\hbox{(Hartogs figure),}\cr
&\wt\Omega=D(R)\times\omega~~~~&\hbox{(filled Hartogs figure).}\cr}$$
where $0\le r<R$ and $D(r)\subset\bC$ denotes the open disk of center $0$
and radius $r$ in~$\bC$.

\vbox{%
\input epsfiles/fig_1_3.tex
\vskip8mm
\centerline{{\bf Fig.~I-3} Hartogs figure}}
\vskip5mm

\noindent
Then every function $f\in\cO(\Omega)$ can be extended to 
$\smash{\wt\Omega}=\omega\times D(R)$ by means of the Cauchy formula:
$$\wt f(z_1,z')={1\over 2\pi\ii}\int_{|\zeta_1|=\rho}
{f(\zeta_1,z')\over\zeta_1-z_1}d\zeta_1,~~~~z\in\wt\Omega,~~~\max\{|z_1|,r\}
<\rho<R.$$
In fact $\wt f\in\cO(D(R)\times\omega)$ and $\wt f=f$ on
$D(R)\times\omega'$, so we must have $\wt f=f$ on $\Omega$ since
$\Omega$ is connected.  It follows that $\Omega$ is not a domain of
holomorphy.  Let us quote two interesting consequences of this example.
\endclaim

\claim{(6.4) Corollary {\rm(Riemann's extension theorem)}} Let $X$ be a
complex analytic manifold, and $S$ a closed submanifold of codimension
$\ge 2$. Then every $f\in\cO(X\ssm S)$ extends holomorphically to $X$.
\endclaim

\proof{Proof} This is a local result. We may choose coordinates $(z_1,\ldots,z_n)$
and a polydisk $D(R)^n$ in the corresponding chart such that 
$S\cap D(R)^n$ is given by equations $z_1=\ldots=z_p=0$, 
$p=\codim S\ge 2$. Then, denoting $\omega=D(R)^{n-1}$ and
$\omega'=\omega\ssm\{z_2=\ldots=z_p=0\}$, the complement $D(R)^n\ssm S$
can be written as the Hartogs figure
$$D(R)^n\ssm S=\big((D(R)\ssm\{0\})\times\omega\big)\cup
\big(D(R)\times\omega'\big).$$
It follows that $f$ can be extended to $\wt\Omega=D(R)^n$.\qed
\endproof

\subsection{6.B}{Holomorphic Convexity and Pseudoconvexity}
Let $X$ be a complex manifold. We first introduce the notion of holomorphic
hull of a compact set $K\subset X$. This can be seen somehow
as the complex analogue of the notion of (affine) convex hull for a compact
set in a real vector space. It is shown that domains of holomorphy in
$\bC^n$ are characterized a property of holomorphic convexity.
Finally, we prove that holomorphic convexity implies pseudoconvexity
-- a complex analogue of the geometric notion of convexity.

\claim{(6.5) Definition} Let $X$ be a complex manifold and let $K$ be
a compact subset of $X$. Then the holomorphic hull of $K$ in $X$ is defined
to be
$$\wh K=\wh K_{\cO(X)}=
\big\{z\in X\,;\,|f(z)|\le\sup_K|f|,~\forall f\in\cO(X)\big\}.$$
\endclaim

\claim{(6.6) Elementary properties} \rm\medskip
\item{a)} $\wh K$ is a closed subset of $X$ containing~$K$.
Moreover we have
$$\sup_{\wh K}|f|=\sup_K|f|,~~~~\forall f\in\cO(X),$$
hence $\wh{\wh K}=\wh K$.
\medskip
\item{b)} If $h:X\to Y$ is a holomorphic map and $K\subset X$ is a
compact set, then $h(\wh K_{\cO(X)})\subset\wh{h(K)}_{\cO(Y)}$.
In particular, if $X\subset Y$, then $\wh K_{\cO(X)}
\subset\wh K_{\cO(Y)}\cap X$. This is immediate from the definition.
\medskip
\item{c)} $\wh K$ contains the union of $K$ with all relatively compact
connected components of $X\ssm K$ (thus $\wh K$ ``fills the holes'' of $K$).
In fact, for every connected component $U$ of $X\ssm K$ we have
$\partial U\subset\partial K$, hence if $\ol U$ is compact
the maximum principle yields
$$\sup_{\ol U}|f|=\sup_{\partial U}|f|\le\sup_{K}|f|,~~~~
\hbox{\rm for all $f\in\cO(X)$}.$$
\item{d)} More generally, suppose that there is a holomorphic map
$h:U\to X$ defined on a relatively compact open set $U$ in a
complex manifold $S$, such that $h$ extends as a continuous map
$h:\ol U\to X$ and $h(\partial U)\subset K$. Then
$h(\ol U)\subset\wh K$. Indeed, for $f\in\cO(X)$, the maximum
principle again yields
$$\sup_{\ol U}|f\circ h|=\sup_{\partial U}|f\circ h|\le\sup_{K}|f|.$$
This is especially useful when $U$ is the unit disk in $\bC$.
\medskip
\item{e)} Suppose that $X=\Omega\subset\bC^n$ is an open set. By
taking $f(z)=\exp(A(z))$ where $A$ is an arbitrary affine function,
we see that $\wh K_{\cO(\Omega)}$ is contained in the intersection of
all affine half-spaces containing~$K$. Hence $\wh K_{\cO(\Omega)}$ is
contained in the affine convex hull $\wh K_\aff$. As a consequence
$\wh K_{\cO(\Omega)}$ is always bounded and $\wh K_{\cO(\bC^n)}$
is a compact set. However, when $\Omega$ is arbitrary,
$\wh K_{\cO(\Omega)}$ is not always compact; for example, in case
$\Omega=\bC^n\ssm\{0\}$, $n\ge 2$, then $\cO(\Omega)=\cO(\bC^n)$
and the holomorphic hull of $K=S(0,1)$ is the non compact set
$\wh K=\ol B(0,1)\ssm\{0\}$.
\vskip0pt
\endclaim

\claim{(6.7) Definition} A complex manifold $X$ is said to be
holomorphically convex if the holomorphic hull $\wh K_{\cO(X)}$
of every compact set $K\subset X$ is compact.
\endclaim

\claim{(6.8) Remark} A complex manifold $X$ is holomorphically
convex if and only if there is an exhausting sequence of holomorphically
compact subsets $K_\nu\subset X$, i.e.\ compact sets such that
$$X=\bigcup K_\nu,~~~~\wh K_\nu=K_\nu,~~~~K^\circ_\nu\supset K_{\nu-1}.$$
{\rm Indeed, if $X$ is holomorphically convex, we may define $K_\nu$
inductively  by $K_0=\emptyset$ and $K_{\nu+1}=(K'_\nu\cup L_\nu)^{\wedge
}_{\cO(X)}$, where $K'_\nu$ is a neighborhood of $K_\nu$ and $L_\nu$ a
sequence of compact sets of $X$ such that $X=\bigcup L_\nu$. The
converse is obvious: if such a sequence $(K_\nu)$ exists, then every
compact subset $K\subset X$ is contained in some $K_\nu$, hence $\wh
K\subset\smash{\wh K}_\nu=K_\nu$ is compact.\qed}
\endclaim

We now concentrate on domains of holomorphy in~$\bC^n$. We denote by $d$
and $B(z,r)$ the distance and the open balls associated to an arbitrary
norm on $\bC^n$, and we set for simplicity $B=B(0,1)$.

\claim{(6.9) Proposition} If $\Omega$ is a domain of holomorphy and
$K\subset\Omega$ is a compact subset, then $d(\wh K,\complement\Omega)=d(K,
\complement\Omega)$ and $\wh K$ is compact.
\endclaim

\proof{Proof} Let $f\in\cO(\Omega)$. Given $r<d(K,\complement\Omega)$, we
denote by $M$ the supremum of $|f|$ on the compact subset $K+r\ol B\subset
\Omega$. Then for every $z\in K$ and $\xi\in\ol B$, the function
$$\bC\ni t\longmapsto f(z+t\xi)=\sum_{k=0}^{+\infty}{1\over k!}
D^kf(z)(\xi)^k\,t^k\leqno(6.10)$$
is analytic in the disk $|t|<r$ and bounded by $M$. The Cauchy inequalities
imply
$$|D^kf(z)(\xi)^k|\le Mk!\,r^{-k},~~~~\forall z\in K,~~~\forall\xi\in\ol B.
$$
As the left hand side is an analytic fuction of $z$ in $\Omega$, the
inequality must also hold for $z\in\smash{\wh K}$, $\xi\in\ol B$. 
Every $f\in\cO(\Omega)$ can thus be extended to any ball $B(z,r)$,
$z\in\smash{\wh K}$, by means of the power series (6.10).  Hence
$B(z,r)$ must be contained in $\Omega$, and this shows that
$d(\smash{\wh K},\complement\Omega)\ge r$.  As
$r<d(K,\complement\Omega)$ was arbitrary, we get $d(\smash{\wh K},
\complement\Omega)\ge d(K,\complement\Omega)$ and the converse
inequality is clear, so $d(\smash{\wh K},\complement\Omega)=
d(K,\complement\Omega)$. As $\smash{\wh K}$ is bounded and closed in 
$\Omega$, this shows that $\smash{\wh K}$ is compact.\qed
\endproof

\claim{(6.11) Theorem} Let $\Omega$ be an open subset of $\bC^n$. The 
following properties are equivalent:
\medskip
\item{\rm a)} $\Omega$ is a domain of holomorphy;
\medskip
\item{\rm b)} $\Omega$ is holomorphically convex;
\medskip
\item{\rm c)} For every countable subset $\{z_j\}_{j\in\bN}\subset
\Omega$ without accumulation points in $\Omega$ and every sequence of
complex numbers $(a_j)$, there exists an interpolation function
$F\in\cO(\Omega)$ such that $F(z_j)=a_j$. 
\medskip
\item{\rm d)} There exists a function $F\in\cO(\Omega)$ which is
unbounded on any neighborhood of any point of $\partial\Omega$.
\vskip0pt
\endclaim

\proof{Proof} d) $\Longrightarrow$ a) is obvious and a) $\Longrightarrow$ b)
is a consequence of Prop.~6.9.
\medskip
\noindent{c)} $\Longrightarrow$ d). If $\Omega=\bC^n$ there is nothing to prove.
Otherwise, select a dense sequence $(\zeta_j)$ in $\partial\Omega$ and
take $z_j\in\Omega$ such that $d(z_j,\zeta_j)<2^{-j}$. Then the
interpolation function $F\in\cO(\Omega)$ such that $F(z_j)=j$ satisfies d).
\medskip
\noindent{b)} $\Longrightarrow$ c). Let $K_\nu\subset\Omega $ be an exhausting
sequence of holomorphically convex compact sets as in Remark~6.8.
Let $\nu(j)$ be the unique index $\nu$ such that
\hbox{$z_j\in K_{\nu(j)+1}\ssm K_{\nu(j)}$}. By the definition of a
holomorphic hull, we can find a function $g_j\in\cO(\Omega)$ such that
$$\sup_{K_{\nu(j)}}|g_j|<|g_j(z_j)|.$$
After multiplying $g_j$ by a constant, we may assume that $g_j(z_j)=1$. Let 
$P_j\in\bC[z_1,\ldots,z_n]$ be a polynomial equal to $1$ at $z_j$ and to $0$ at
$z_0,z_1,\ldots,z_{j-1}$. We set
$$F=\sum_{j=0}^{+\infty}\lambda_jP_jg_j^{m_j},$$
where $\lambda_j\in\bC$ and $m_j\in\bN$ are chosen inductively such that
$$\eqalign{
\lambda_j&=a_j-\sum_{0\le k<j}\lambda_k P_k(z_j) g_k(z_j)^{m_k},\cr
|\lambda_j&P_jg_j^{m_j}|\le 2^{-j}~~~~\hbox{\rm on}~~K_{\nu(j)}~;\cr}$$
once $\lambda_j$ has been chosen, the second condition holds as soon as
$m_j$ is large enough. Since $\{z_j\}$ has no accumulation point in
$\Omega$, the sequence $\nu(j)$ tends to $+\infty$, hence the series 
converges uniformly on compact sets.\qed
\endproof

We now show that a holomorphically convex manifold must satisfy some
more geometric convexity condition, known as pseudoconvexity, which is
most easily described in terms of the existence of plurisubharmonic
exhaustion functions.

\claim{(6.12) Definition} A function $\psi:X\longrightarrow[-\infty,+\infty[$ on
a topological space $X$ is said to be an exhaustion if all sublevel
sets $X_c:=\{z\in X\,;\,\psi(z)<c\}$, $c\in\bR$, are relatively compact.
Equivalently, $\psi$ is an exhaustion if and only if $\psi$ tends to
$+\infty$ relatively to the filter of complements $X\ssm K$ of compact
subsets of~$X$.
\endclaim

A function $\psi$ on an open set $\Omega\subset\bR^n$ is thus
an exhaustion if and only if $\psi(x)\to+\infty$ as $x\to\partial\Omega$
or $x\to\infty\,$. It is easy to check, cf.\ Exercise~8.8, that a 
connected open set $\Omega\subset\bR^n$ is convex if and only if $\Omega$
has a locally convex exhaustion function. Since plurisubharmonic
functions appear as the natural generalization of convex functions
in complex analysis, we are led to the following definition.

\claim{(6.13) Definition} Let $X$ be a complex $n$-dimensional manifold.
Then $X$ is said to be
\medskip
\item{\rm a)} weakly pseudoconvex if there exists a smooth
plurisubharmonic exhaustion function $\psi\in\Psh(X)\cap \cC^\infty(X)\,;$
\smallskip
\item{\rm b)} strongly pseudoconvex if there exists a smooth strictly
plurisubharmonic exhaustion function $\psi\in\Psh(X)\cap \cC^\infty(X)$,
i.e.\ $H\psi$ is positive definite at every point.
\vskip0pt
\endclaim

\claim{(6.14) Theorem} Every holomorphically convex manifold $X$ is
weakly pseudoconvex.
\endclaim

\proof{Proof} Let $(K_\nu)$ be an exhausting sequence of holomorphically
convex compact sets as in Remark~6.8.
For every point $a\in L_\nu:=K_{\nu+2}\ssm K^\circ_{\nu+1}$, one
can select $g_{\nu,a}\in\cO(\Omega)$ such that $\sup_{K_\nu}|g_{\nu,a}|<1$
and $|g_{\nu,a}(a)|>1$.  Then $|g_{\nu,a}(z)|>1$ in a neighborhood of $a$~;
by the Borel-Lebesgue lemma, one can find finitely many functions
$(g_{\nu,a})_{a\in I_\nu}$ such that
$$\max_{a\in I_\nu}\big\{|g_{\nu,a}(z)|\big\}>1~~\hbox{\rm for}~~z\in
L_\nu,~~~~\max_{a\in I_\nu}\big\{|g_{\nu,a}(z)|\big\}<1
~~\hbox{\rm for}~~z\in K_\nu.$$
For a sufficiently large exponent $p(\nu)$ we get
$$\sum_{a\in I_\nu}|g_{\nu,a}|^{2p(\nu)}\ge\nu~~\hbox{\rm on}~~L_\nu,~~~~
\sum_{a\in I_\nu}|g_{\nu,a}|^{2p(\nu)}\le 2^{-\nu}~~\hbox{\rm on}~~K_\nu.$$
It follows that the series 
$$\psi(z)=\sum_{\nu\in\bN}\sum_{a\in I_\nu}|g_{\nu,a}(z)|^{2p(\nu)}$$
converges uniformly to a real analytic function $\psi\in\Psh(X)$
(see Exercise~8.11). By construction $\psi(z)\ge\nu$ for $z\in L_\nu$, hence
$\psi$ is an exhaustion.\qed
\endproof

\claim{(6.15) Example} \rm The converse to Theorem~6.14 does not hold.
In fact let $X=\bC^2/\Gamma$ be the quotient of $\bC^2$ by the free
abelian group of rank $2$ generated by the affine automorphisms
$$g_1(z,w)=(z+1,e^{\ii\theta_1}w),~~~~g_2(z,w)=(z+\ii,e^{\ii\theta_2}w),~~~~
\theta_1,\,\theta_2\in\bR.$$
Since $\Gamma$ acts properly discontinuously on $\bC^2$, the quotient
has a structure of a complex (non compact) $2$-dimensional manifold.
The function $w\mapsto |w|^2$ is $\Gamma$-invariant, hence it induces a
function $\psi((z,w)^\sim)=|w|^2$ on~$X$ which is in fact a
plurisubharmonic exhaustion function. Therefore $X$ is weakly
pseudoconvex. On the other hand, any holomorphic function $f\in\cO(X)$
corresponds to a $\Gamma$-invariant holomorphic function $\smash{\wt
f}(z,w)$ on~$\bC^2$. Then $z\mapsto\wt f(z,w)$ is bounded for $w$
fixed, because $\wt f(z,w)$ lies in the image of the compact set
$K\times\ol D(0,|w|)$, $K={}$ unit square in $\bC$. By Liouville's
theorem, $\wt f(z,w)$ does not depend on $z$. Hence functions
$f\in\cO(X)$ are in one-to-one correspondence with holomorphic
functions $\wt f(w)$ on $\bC$ such that $\smash{\wt
f}(e^{\ii\theta_j}w)= \smash{\wt f}(w)$. By looking at the Taylor
expansion at the origin, we conclude that $\smash{\wt f}$ must be a
constant if $\theta_1\notin\bQ$ or $\theta_2\notin\bQ$ (if
$\theta_1,\theta_2\in\bQ$ and $m$ is the least common denominator of
$\theta_1,\theta_2$, then ${\wt f}$ is a power series of the form
$\sum\alpha_kw^{mk}$). From this, it follows easily that $X$ is
holomorphically convex if and only if $\theta_1,\theta_2\in\bQ$.
\endclaim

\subsection{6.C}{Stein Manifolds}
The class of holomorphically convex manifolds contains two types of
manifolds of a rather different nature:
\smallskip\noindent
$\bu$ domains of holomorphy $X=\Omega\subset\bC^n\,;$
\smallskip\noindent
$\bu$ compact complex manifolds.
\smallskip\noindent
In the first case we have a lot of holomorphic functions, in fact
the functions in $\cO(\Omega)$ separate any pair of points of~$\Omega$.
On the other hand, if $X$ is compact and connected, the sets
$\Psh(X)$ and $\cO(X)$ consist of constant functions merely (by the
maximum principle). It is therefore desirable to introduce a clear
distinction between these two subclasses. For this purpose, 
\cite{Stein 1951}
introduced the class of manifolds which are now called Stein manifolds.

\claim{(6.16) Definition} A complex manifold $X$ is said to be a Stein
manifold if
\medskip
\item{\rm a)} $X$ is holomorphically convex\/$;$
\smallskip
\item{\rm b)} $\cO(X)$ locally separates points in $X$, i.e.\ every
point $x\in X$ has a neighborhood $V$ such that for any $y\in V\ssm\{x\}$
there exists $f\in\cO(X)$ with $f(y)\ne f(x)$.
\vskip0pt
\endclaim
The second condition is automatic if $X=\Omega$ is an open subset of
$\bC^n$. Hence an open set $\Omega\subset\bC^n$ is Stein if and only
if $\Omega$ is a domain of holomorphy.

\claim{(6.17) Lemma} If a complex manifold $X$ satisfies the axiom
$(6.16~{\rm b})$ of local separation, there exists a smooth
nonnegative strictly plurisubharmonic function $u\in\Psh(X)$.
\endclaim

\proof{Proof} Fix $x_0\in X$. We first show that there exists a smooth
nonnegative function $u_0\in\Psh(X)$ which is strictly plurisubharmonic
on a neighborhood of~$x_0$. Let $(z_1,\ldots,z_n)$ be local analytic
coordinates centered at $x_0$, and if necessary, replace $z_j$ by
$\lambda z_j$ so that the closed unit ball $\ol B=\{\sum|z_j|^2\le 1\}$
is contained in the neighborhood $V\ni x_0$ on which (6.16~b) holds.
Then, for every point $y\in\partial B$, there exists a holomorphic
function $f\in\cO(X)$ such that $f(y)\ne f(x_0)$. Replacing $f$
with $\lambda(f-f(x_0))$, we can achieve \hbox{$f(x_0)=0$} and $|f(y)|>1$.
By compactness of $\partial B$, we find finitely many functions
$f_1,\ldots,f_N\in\cO(X)$ such that $v_0=\sum|f_j|^2$ satisfies $v_0(x_0)=0$,
while $v_0\ge 1$ on~$\partial B$. Now, we set
$$u_0(z)=\cases{
v_0(z)&on $X\ssm B$,\cr
M_\varepsilon\{v_0(z),(|z|^2+1)/3\}&on $B$.\cr}$$
where $M_\varepsilon$ are the regularized max functions defined in~5.18.
Then $u_0$ is smooth and plurisubharmonic, coincides with $v_0$ near
$\partial B$ and with $(|z|^2+1)/3$ on a neighborhood
of~$x_0$. We can cover $X$ by countably many neighborhoods~$(V_j)_{j\ge 1}$,
for which we have a smooth plurisubharmonic functions $u_j\in\Psh(X)$
such that $u_j$ is strictly plurisubharmonic on~$V_j$. Then select a
sequence $\varepsilon_j>0$ converging to $0$ so fast that
$u=\sum\varepsilon_ju_j\in \cC^\infty(X)$. The function $u$ is nonnegative
and strictly plurisubharmonic everywhere on~$X$.\qed
\endproof

\claim{(6.18) Theorem} Every Stein manifold is strongly pseudoconvex.
\endclaim

\proof{Proof} By Th.~6.14, there is a smooth exhaustion function
$\psi\in\Psh(X)$. If \hbox{$u\ge 0$} is strictly plurisubharmonic, then
$\psi'=\psi+u$ is a strictly plurisubharmonic exhaustion.\qed
\endproof

The converse problem to know whether every strongly pseudoconvex manifold
is actually a Stein manifold is known as the {\it Levi problem}, and was
raised by \cite{Levi 1910} in the case of domains $\Omega\subset\bC^n$. In 
that case, the problem has been solved in the affirmative independently by
\cite{Oka 1953}, \cite{Norguet 1954} and \cite{Bremermann 1954}. 
The general solution of the Levi problem has been obtained
by \cite{Grauert 1958}. Our proof will rely on the theory of $L^2$
estimates for $d''$, which will be available only in Chapter~VIII.

\vbox{%
\input epsfiles/fig_1_4.tex
\vskip6mm
\centerline{{\bf Fig.~I-4} Hartogs figure with excrescence}}
\vskip5mm

\claim{(6.19) Remark} \rm It will be shown later that Stein manifolds
always have enough holomorphic functions to separate finitely many points,
and one can even interpolate given values of a function and its derivatives
of some fixed order at any discrete set of points. In particular, we might
have replaced condition (6.16~b) by the stronger requirement that $\cO(X)$
separates any pair of points. On the other hand, there are examples of
manifolds satisfying the local separation condition (6.16~b), but not
global separation. A simple example is obtained by attaching an excrescence
inside a Hartogs figure, in such a way that the resulting map
$\pi:X\to D=D(0,1)^2$ is not one-to-one (see Figure~I-4 above); then
$\cO(X)$ coincides with $\pi^\star\cO(D)$.
\endclaim

\subsection{6.D}{Heredity Properties}
Holomorphic convexity and pseudoconvexity are preserved under quite
a number of natural constructions. The main heredity properties can
be summarized in the following Proposition.

\claim{(6.20) Proposition} Let $\cC$ denote the class of
holomorphically convex $($resp.\ of Stein, or weakly pseudoconvex,
strongly pseudoconvex manifolds$)$.
\medskip
\item{\rm a)} If $X,Y\in\cC$, then $X\times Y\in\cC$.
\smallskip
\item{\rm b)} If $X\in\cC$ and $S$ is a closed complex submanifold of $X$,
then $S\in\cC$.
\smallskip
\item{\rm c)} If $(S_j)_{1\le j\le N}$ is a collection of $($not
necessarily closed$)$ submanifolds of a complex manifold~$X$ such that
$S=\bigcap S_j$ is a submanifold of~$X$, and if $S_j\in\cC$ for all~$j$,
then $S\in\cC$.
\smallskip
\item{\rm d)} If $F:X\to Y$ is a holomorphic map and $S\subset X$,
$S'\subset Y$ are $($not necessarily closed$)$ submanifolds in the
class $\cC$, then $S\cap F^{-1}(S')$ is in~$\cC$, as long as it is
a submanifold of~$X$.
\smallskip
\item{\rm e)} If $X$ is a weakly $($resp.\ strongly$)$ pseudoconvex
manifold and $u$ is a smooth plurisubharmonic function on~$X$, then
the open set $\Omega=u^{-1}(]-\infty,c[)$ is weakly $($resp.\ strongly$)$
pseudoconvex. In particular the sublevel sets
$$X_c=\psi^{-1}(]-\infty,c[)$$
of a $($strictly$)$ plurisubharmonic exhaustion function are
weakly $($resp.\ strongly$)$ pseudoconvex.
\vskip0pt
\endclaim

\proof{Proof} All properties are more or less immediate to check, so we only
give the main facts.
\medskip\noindent
a) For $K\subset X$, $L\subset Y$ compact, we have
$(K\times L)^{\wedge}_{\cO(X\times Y)}=\hbox{$\wh K_{\cO(X)}\times
\wh K_{\cO(Y)}$}$, and if $\varphi$, $\psi$ are plurisubharmonic exhaustions
of $X$, $Y$, then \hbox{$\varphi(x)+\psi(y)$} is a plurisubharmonic
exhaustion of~$X\times Y$.
\medskip\noindent
b) For a compact set $K\subset S$, we have $\wh K_{\cO(S)}\subset
\wh K_{\cO(X)}\cap S$, and if $\psi\in\Psh(X)$ is an exhaustion, then
$\psi_{\restriction S}\in\Psh(S)$ is an exhaustion (since $S$ is closed).
\medskip\noindent
c) $\bigcap S_j$ is a closed submanifold in $\prod S_j$ (equal to its
intersection with the diagonal of~$X^N$).
\medskip\noindent
d) For a compact set $K\subset S\cap F^{-1}(S')$, we have
$$\wh K_{\cO(S\cap F^{-1}(S'))}\subset\wh K_{\cO(S)}\cap
F^{-1}(\wh{F(K)}_{\cO(S')}),$$
and if $\varphi$, $\psi$ are plurisubharmonic exhaustions of $S$, $S'$,
then $\varphi+\psi\circ F$ is a plurisubharmonic exhaustion of
$S\cap F^{-1}(S')$.
\medskip\noindent
e) $\varphi(z):=\psi(z)+1/(c-u(z))$ is a (strictly) plurisubharmonic
exhaustion function on~$\Omega$.\qed
\endproof

\section{7}{Pseudoconvex Open Sets in $\bC^n$}
\subsection{7.A}{Geometric Characterizations of Pseudoconvex Open Sets}
We first discuss some characterizations of pseudoconvex open sets
in~$\bC^n$. We will need the following elementary criterion for
plurisubharmonicity.

\claim{(7.1) Criterion} Let $v:\Omega\longrightarrow[-\infty,+\infty[$ be an
upper semicontinuous function. Then $v$ is plurisubharmonic if and only if for 
every closed disk $\ol\Delta=z_0+\ol D(1)\eta\subset\Omega$ and every 
polynomial $P\in\bC[t]$ such that $v(z_0+t\eta)\le\Re P(t)$ for $|t|=1$,
then $v(z_0)\le\Re P(0)$.
\endclaim

\proof{Proof} The condition is necessary because $t\longmapsto v(z_0+t\eta)-\Re
P(t)$ is subharmonic in a neighborhood of $\ol D(1)$, so it satisfies the
maximum principle on $D(1)$ by Th.~4.14. Let us prove now the 
sufficiency. The upper semicontinuity of $v$ implies $v=\lim_{\nu\to+\infty}
v_\nu$ on $\partial\Delta$ where $(v_\nu)$ is a strictly decreasing sequence of
continuous functions on $\partial\Delta$. As trigonometric polynomials are dense
in $C^0(S^1,\bR)$, we may assume $v_\nu(z_0+e^{\ii\theta}\eta)=\Re P_\nu
(e^{\ii\theta})$, $P_\nu\in\bC[t]$. Then $v(z_0+t\eta)\le\Re P_\nu(t)$ for
$|t|=1$, and the hypothesis implies
$$v(z_0)\le\Re P_\nu(0)={1\over 2\pi}\int_0^{2\pi}\Re P_\nu(e^{\ii\theta})\,
d\theta={1\over 2\pi}\int_0^{2\pi}v_\nu(z_0+e^{\ii\theta}\eta)\,d\theta.$$
Taking the limit when $\nu$ tends to $+\infty$ shows that $v$ satisfies the
mean value inequality (5.2).\qed
\endproof

For any $z\in\Omega$ and $\xi\in\bC^n$, we denote by
$$\delta_\Omega(z,\xi)=\sup\big\{r>0~;~z+D(r)\,\xi\subset\Omega\big\}$$
the distance from $z$ to $\partial\Omega$ in the complex direction $\xi$.

\claim{(7.2) Theorem} Let $\Omega\subset\bC^n$ be an open subset. The
following properties are equivalent:
\medskip
\item{\rm a)} $\Omega$ is strongly pseudoconvex $($according to
Def.~6.13~{\rm b)}$;$
\smallskip
\item{\rm b)} $\Omega$ is weakly pseudoconvex\/$;$
\smallskip
\item{\rm c)} $\Omega$ has a plurisubharmonic exhaustion function $\psi$.
\smallskip
\item{\rm d)} $-\log\delta_\Omega(z,\xi)$ is plurisubharmonic on 
$\Omega\times\bC^n~;$
\smallskip
\item{\rm e)} $-\log d(z,\complement\Omega)$ is plurisubharmonic on $\Omega$.
\medskip\noindent
If one of these properties hold, $\Omega$ is said to be a pseudoconvex
open set.
\endclaim

\proof{Proof} The implications a) $\Longrightarrow$ b) $\Longrightarrow$ c)
are obvious. For the implication c) $\Longrightarrow$ d), we use Criterion 7.1.
Consider a disk $\ol\Delta=(z_0,\xi_0)+\ol D(1)\,(\eta,\alpha)$ in 
$\Omega\times\bC^n$ and a polynomial $P\in\bC[t]$ such that
$$-\log\delta_\Omega(z_0+t\eta,\xi_0+t\alpha)\le\Re P(t)~~~
\hbox{\rm for}~~|t|=1.$$
We have to verify that the inequality also holds when $|t|<1$. Consider
the holomorphic mapping $h:\bC^2\longrightarrow\bC^n$ defined by
$$h(t,w)=z_0+t\eta+we^{-P(t)}(\xi_0+t\alpha).$$
By hypothesis
$$\eqalign{
&h\big(\ol D(1)\times\{0\}\big)={\rm pr}_1(\ol\Delta)\subset\Omega,\cr
&h\big(\partial D(1)\times D(1)\big)\subset\Omega~~~\hbox{\rm(since}~
|e^{-P}|\le\delta_\Omega~~\hbox{\rm on}~\partial\Delta),\cr}$$
and the desired conclusion is that $h\big(\ol D(1)\times D(1)\big)
\subset\Omega$. Let $J$ be the set of radii $r\ge 0$ such that
$h\big(\ol D(1)\times\ol D(r)\big)\subset\Omega$. Then $J$ is an
open interval $[0,R[$, $R>0$. If $R<1$, we get a
contradiction as follows. Let $\psi\in\Psh(\Omega)$ be an exhaustion function
and
$$K=h\big(\partial D(1)\times\ol D(R)\big)\compact\Omega,~~~~
c=\sup_K\psi.$$
As $\psi\circ h$ is plurisubharmonic on a neighborhood of $\ol D(1)\times
D(R)$, the maximum principle applied with respect to $t$ implies
$$\psi\circ h(t,w)\le c~~~\hbox{\rm on}~~\ol D(1)\times D(R),$$
hence $h\big(\ol D(1)\times D(R)\big)\subset\Omega_c\compact\Omega$
and $h\big(\ol D(1)\times\ol D(R+\varepsilon)\big)\subset\Omega$
for some $\varepsilon>0$, a contradiction.
\medskip
\noindent{d)} $\Longrightarrow$ e). The function~ $-\log d(z,\complement
\Omega)$ is continuous on $\Omega$ and satisfies the mean value inequality 
because
$$-\log d(z,\complement\Omega)=\sup_{\xi\in\ol B}\big(-\log\delta_\Omega(z,\xi)
\big).$$
\medskip
\noindent{e)} $\Longrightarrow$ a). It is clear that
$$u(z)=|z|^2+\max\{\log d(z,\complement\Omega)^{-1},0\}$$
is a continuous strictly plurisubharmonic exhaustion function. Richberg's 
theorem 5.21 implies that there exists $\psi\in \cC^\infty(\Omega)$ strictly
plurisubharmonic such that $u\le\psi\le u+1$. Then $\psi$ is the required
exhaustion function.\qed
\endproof

\claim{(7.3) Proposition} \smallskip
\item{\rm a)} Let $\Omega\subset\bC^n$ and
$\Omega'\subset\bC^p$ be pseudoconvex. Then $\Omega\times\Omega'$ is
pseudoconvex. For every holomorphic map $F:\Omega\to\bC^p$ the
inverse image $F^{-1}(\Omega')$ is pseudoconvex.
\smallskip
\item{\rm b)} If $(\Omega_\alpha)_{\alpha\in I}$ is a family
of pseudoconvex open subsets of $\bC^n$, the interior of the
intersection $\Omega=\big(\bigcap_{\alpha\in I}
\Omega_\alpha\big)^\circ$ is pseudoconvex.
\smallskip
\item{\rm c)} If $(\Omega_j)_{j\in\bN}$ is a non decreasing sequence
of pseudoconvex open subsets of $\bC^n$, then 
$\Omega=\bigcup_{j\in\bN}\Omega_j$ is pseudoconvex.
\vskip0pt
\endclaim

\proof{Proof} a) Let $\varphi,\psi$ be smooth plurisubharmonic exhaustions
of $\Omega,\Omega'$. Then $(z,w)\longmapsto\varphi(z)+\psi(w)$ is an
exhaustion of $\Omega\times\Omega'$ and $z\longmapsto\varphi(z)+
\psi(F(z))$ is an exhaustion of $F^{-1}(\Omega')$.
\medskip
\noindent{b)} We have $-\log d(z,\complement\Omega)=\sup_{\alpha\in I}
-\log d(z,\complement\Omega_\alpha)$, so this function
is pluri\-sub\-harmonic.
\medskip
\noindent{c)}
The limit $-\log d(z,\complement\Omega)={{\lim}{\downarrow}\,}_{j\to+\infty}
-\log d(z,\complement\Omega_j)$ is plurisubharmonic, hence $\Omega$ is
pseudoconvex. This result cannot be generalized to strongly pseudoconvex
manifolds: J.E.~Fornaess in \cite{Fornaess 1977} has constructed an 
increasing sequence of $2$-dimensional Stein (even affine algebraic)
manifolds $X_\nu$ whose union is not Stein; see Exercise~8.16.\qed
\endproof

\claim{(7.4) Examples} \smallskip\rm
\noindent{a)} An {\it analytic polyhedron} in $\bC^n$ is
an open subset of the form
$$P=\{z\in\bC^n\,;\,|f_j(z)|<1,~1\le j\le N\}$$
where $(f_j)_{1\le j\le N}$ is a family of analytic functions on
$\bC^n$. By 7.3~a), every analytic polyhedron is pseudoconvex.
\medskip
\noindent{b)} Let $\omega\subset\bC^{n-1}$ be pseudoconvex and let
$u:\omega\longrightarrow[-\infty,+\infty[$ be an upper semicontinuous function. Then the 
{\it Hartogs domain}
$$\Omega=\big\{(z_1,z')\in\bC\times\omega\,;\,\log|z_1|+u(z')<0\big\}$$
is pseudoconvex if and and only if $u$ is plurisubharmonic.
To see that the plurisubharmonicity of $u$ is necessary, observe that 
$$u(z')=-\log\delta_\Omega\big((0,z'),(1,0)\big).$$
Conversely, assume that $u$ is plurisubharmonic and continuous. If
$\psi$ is a plurisubharmonic exhaustion of $\omega$, then
$$\psi(z')+\big|\log|z_1|+u(z')\big|^{-1}$$
is an exhaustion of $\Omega$. This is no longer true if $u$ is not 
continuous, but in this case we may apply Property 7.3~c) to
conclude that
$$\Omega_\varepsilon=\big\{(z_1,z')\,;\,d(z',\complement\omega)>
\varepsilon,~\log|z_1|+u\star\rho_\varepsilon(z')<0\big\},~~~~
\Omega=\bigcup\Omega_\varepsilon$$
are pseudoconvex.
\medskip
\noindent{c)} An open set $\Omega\subset\bC^n$ is called
a {\it tube} of base $\omega$ if 
$\Omega=\omega+\ii\bR^n$ for some open subset $\omega\subset\bR^n$.
Then of course $-\log d(z,\complement\Omega)=-\log(x,\complement\omega)$
depends only on the real part $x=\Re z$. By Th.~5.13, this
function is plurisubharmonic if and only if it is locally convex in $x$.
Therefore $\Omega$ if pseudoconvex if and only if every connected
component of $\omega$ is convex.
\medskip
\noindent{d)} An open set $\Omega\subset\bC^n$ is called a {\it Reinhardt 
domain} if $(e^{\ii\theta_1}z_1,\ldots,e^{\ii\theta_n}z_n)$ is in $\Omega$ for
every $z=(z_1,\ldots,z_n)\in\Omega$ and $\theta_1,\ldots,\theta_n\in\bR^n$.
For such a domain, we consider the {\it logarithmic indicatrix}
$$\omega^\star=\Omega^\star\cap\bR^n~~~\hbox{\rm with}~~~
\Omega^\star=\{\zeta\in\bC^n\,;\,(e^{\zeta_1},\ldots,e^{\zeta_n})\in\Omega\}.$$
It is clear that $\Omega^\star$ is a tube of base $\omega^\star$.
Therefore every connected component of $\omega^\star$ must be convex
if $\Omega$ is pseudoconvex. 
The converse is not true: $\Omega=\bC^n\ssm\{0\}$ is not pseudoconvex
for $n\ge 2$ although $\omega^\star=\bR^n$ is convex. However, the 
Reinhardt open set
$$\Omega^\bullet=\big\{(z_1,\ldots,z_n)\in(\bC\ssm\{0\})^n\,;\,
(\log|z_1|,\ldots,\log|z_n|)\in\omega^\star\big\}\subset\Omega$$
is easily seen to be pseudoconvex if $\omega^\star$ is convex:
if $\chi$ is a convex exhaustion of $\omega^\star$, then
$\psi(z)=\chi(\log|z_1|,\ldots,\log|z_n|)$ is a plurisubharmonic
exhaustion of $\Omega^\bullet$. Similarly, if $\omega^\star$ is 
convex and such that $x\in\omega^\star \Longrightarrow y\in\omega^\star$
for $y_j\le x_j$, we can take $\chi$ increasing in all variables and
tending to $+\infty$ on $\partial\omega^\star$, hence the set
$$\wt\Omega=\big\{(z_1,\ldots,z_n)\in\bC^n\,;\,|z_j|\le e^{x_j}~~
\hbox{\rm for~some~~}x\in\omega^\star\big\}$$
is a pseudoconvex Reinhardt open set containing $0$.\qed
\endclaim

\subsection{7.B}{Kiselman's Minimum Principle} 
We already know that a maximum of plurisubharmonic functions is
plurisubharmonic.  However, if $v$ is a plurisubharmonic function on
$X\times\bC^n$, the partial minimum function on $X$ defined by
$u(\zeta)=\inf_{z\in\Omega}v(\zeta,z)$ need not be plurisubharmonic.  
A simple counterexample in $\bC\times\bC$ is given by
$$v(\zeta,z)=|z|^2+2\Re(z\zeta)=|z+\ol\zeta|^2-|\zeta|^2,~~~~
u(\zeta)=-|\zeta|^2.$$
It follows that the image $F(\Omega)$ of a pseudoconvex open set
$\Omega$ by a holomorphic map $F$ need not be pseudoconvex. In fact, if
  $$\Omega=\{(t,\zeta,z)\in\bC^3\,;\,\log|t|+v(\zeta,z)<0\}$$
and if $\Omega'\subset\bC^2$ is the image of $\Omega$ by
the projection map $(t,\zeta,z)\longmapsto(t,\zeta)$,
then $\Omega'=\{(t,\zeta)\in\bC^2\,;\,\log|t|+u(\zeta)<0\}$ is
not pseudoconvex. However, the minimum property holds true when
$v(\zeta,z)$ depends only on $\Re z$~:

\claim{(7.5) Theorem {\rm (\cite{Kiselman 1978})}} Let 
$\Omega\subset\bC^p\times\bC^n$ be a pseudoconvex open set such that
each slice
$$\Omega_\zeta=\{z\in\bC^n\,;\,(\zeta,z)\in\Omega\},~~~~\zeta\in\bC^p,$$
is a convex tube $\omega_\zeta+\ii\bR^n$, $\omega_\zeta\subset\bR^n$.
For every plurisubharmonic function $v(\zeta,z)$ on $\Omega$ that
does not depend on $\Im z$, the function
$$u(\zeta)=\inf_{z\in\Omega_\zeta}~v(\zeta,z)$$
is plurisubharmonic or locally $\equiv-\infty$ on $\Omega'=
{\rm pr}_{\bC^n}(\Omega)$.
\endclaim

\proof{Proof} The hypothesis implies that $v(\zeta,z)$ is
convex in $x=\Re z$. In addition, we first assume that
$v$ is smooth, plurisubharmonic in $(\zeta,z)$, strictly convex in $x$
and $\lim_{x\to\{\infty\}\cup\partial\omega_\zeta}v(\zeta,x)=+\infty$ 
for every $\zeta\in\Omega'$.
Then $x\longmapsto v(\zeta,x)$ has a unique minimum point $x=g(\zeta)$,
solution of the equations $\partial v/\partial x_j(x,\zeta)=0$.
As the matrix $(\partial^2 v/\partial x_j\partial x_k)$ is positive 
definite, the implicit function theorem shows that $g$ is smooth.
Now, if $\bC\ni w\longmapsto\zeta_0+wa$, $a\in\bC^n$, $|w|\le 1$ is
a complex disk $\Delta$ contained in $\Omega$, there exists a holomorphic
function $f$ on the unit disk, smooth up to the boundary, whose
real part solves the Dirichlet problem
$$\Re f(e^{\ii\theta})=g(\zeta_0+e^{\ii\theta}a).$$
Since $v(\zeta_0+wa,f(w))$ is subharmonic in $w$, we get the
mean value inequality
$$v(\zeta_0,f(0))\le{1\over 2\pi}\int_0^{2\pi}v\big(\zeta_0+e^{\ii\theta}a,
f(e^{\ii\theta})\big)d\theta=
{1\over 2\pi}\int_{\partial\Delta}v(\zeta,g(\zeta))d\theta.$$
The last equality holds because $\Re f=g$ on $\partial\Delta$ and
$v(\zeta,z)=v(\zeta,\Re z)$ by hypothesis.
As $u(\zeta_0)\le v(\zeta_0,f(0))$ and $u(\zeta)=v(\zeta,g(\zeta))$,
we see that $u$ satisfies the mean value inequality, thus $u$ is
plurisubharmonic.

Now, this result can be extended to arbitrary functions $v$ as follows:
let $\psi(\zeta,z)\ge 0$ be a continuous plurisubharmonic function on
$\Omega$ which is independent of $\Im z$ and is an exhaustion of
$\Omega\cap(\bC^p\times\bR^n)$, e.g.\
$$\psi(\zeta,z)=\max\{|\zeta|^2+|\Re z|^2,-\log\delta_\Omega(\zeta,z)\}.$$
There is slowly increasing sequence $C_j\to+\infty$ such that each
function $\psi_j=(C_j-\psi\star\rho_{1/j})^{-1}$ is an ``exhaustion"
of a pseudoconvex open set $\Omega_j\compact\Omega$ whose slices are convex
tubes and such that $d(\Omega_j,\complement\Omega)>2/j$. Then
$$v_j(\zeta,z)=v\star\rho_{1/j}(\zeta,z)+{1\over j}|\Re z|^2+\psi_j(\zeta,z)$$
is a decreasing sequence of plurisubharmonic functions on $\Omega_j$
satisfying our previous conditions. As $v=\lim v_j$, we see that
$u=\lim u_j$ is plurisubharmonic.\qed
\endproof

\claim{(7.6) Corollary} Let $\Omega\subset\bC^p\times\bC^n$ be a 
pseudoconvex open set such that all slices $\Omega_\zeta$,
$\zeta\in\bC^p$, are convex tubes in $\bC^n$. Then the projection 
$\Omega'$ of $\Omega$ on $\bC^p$ is pseudoconvex.
\endclaim

\proof{Proof} Take $v\in\Psh(\Omega)$ equal to the function $\psi$
defined in the proof of Th.~7.5. Then $u$ is a 
plurisubharmonic exhaustion of $\Omega'$.\qed
\endproof

\subsection{7.C}{Levi Form of the Boundary}
For an arbitrary domain in $\bC^n$, we first show that pseudoconvexity is a 
local property of the boundary.

\claim{(7.7) Theorem} Let $\Omega\subset\bC^n$ be an open subset such that
every point $z_0\in\partial\Omega$ has a neighborhood $V$ such that
$\Omega\cap V$ is pseudoconvex. Then $\Omega$ is pseudoconvex.
\endclaim

\proof{Proof} As $d(z,\complement\Omega)$ coincides with $d\big(z,\complement
(\Omega\cap V)\big)$ in a neighborhood of $z_0$, we see that there exists a 
neighborhood $U$ of $\partial\Omega$ such that $-\log d(z,\complement\Omega)$
is plurisubharmonic on $\Omega\cap U$. Choose a convex increasing function
$\chi$ such that
$$\chi(r)>\sup_{(\Omega\ssm U)\cap\ol B(0,r)}-\log d(z,\complement
\Omega),~~~~\forall r\ge 0.$$
Then the function
$$\psi(z)=\max\big\{\chi(|z|),-\log d(z,\complement\Omega)\big\}$$
coincides with $\chi(|z|)$ in a neighborhood of $\Omega\ssm U$. 
Therefore $\psi\in\Psh(\Omega)$, and $\psi$ is clearly an exhaustion.\qed
\endproof

Now, we give a geometric characterization of the pseudoconvexity property when
$\partial\Omega$ is of class $C^2$. Let $\rho\in C^2(\ol\Omega)$ be a 
{\it defining function} of $\Omega$, i.e.\ a function such that
$$\rho<0~~\hbox{\rm on}~~\Omega,~~~~\rho=0~~\hbox{\rm and}~~d\rho\ne 0~~
\hbox{\rm on}~~\partial\Omega.\leqno(7.9)$$
The {\it holomorphic tangent space} to $\partial\Omega$ is by definition the
largest complex subspace which is contained in the tangent space
$T_{\partial\Omega}$ to the boundary:
$${}^hT_{\partial\Omega}=T_{\partial\Omega}\cap JT_{\partial\Omega}.
\leqno(7.9)$$
It is easy to see that ${}^hT_{\partial\Omega,z}$ is the complex
hyperplane of  vectors $\xi\in\bC^n$ such that 
$$d'\rho(z)\cdot\xi=
\sum_{1\le j\le n}{\partial\rho\over\partial z_j}\,\xi_j=0.$$
The {\it Levi form} on ${}^hT_{\partial\Omega}$ is defined at every point
$z\in\partial\Omega$ by
$$L_{\partial\Omega,z}(\xi)={1\over|\nabla\rho(z)|}\sum_{j,k}
{\partial^2\rho\over\partial z_j\partial\ol z_k}\,\xi_j\ol\xi_k,~~~~
\xi\in {}^hT_{\partial\Omega,z}.\leqno(7.10)$$
The Levi form does not depend on the particular choice of $\rho$, as can be 
seen from the following intrinsic computation of $L_{\partial\Omega}$
(we still denote by $L_{\partial\Omega}$ the associated sesquilinear form).

\claim{(7.11) Lemma} Let $\xi,\eta$ be $C^1$ vector fields on
$\partial\Omega$ with values in ${}^hT_{\partial\Omega}$. Then
$$\langle[\xi,\eta],J\nu\rangle=4\Im L_{\partial\Omega}(\xi,\eta)$$
where $\nu$ is the outward normal unit vector to $\partial\Omega$,
$[~,~]$ the Lie bracket of vector fields and $\langle~,~\rangle$ the 
hermitian inner product.
\endclaim

\proof{Proof} Extend first $\xi,\eta$ as vector fields in a neighborhood of
$\partial\Omega$ and set
$$\xi'=\sum\xi_j{\partial\over\partial z_j}={1\over 2}(\xi-\ii J\xi),~~~~
\eta''=\sum\ol\eta_k{\partial\over\partial\ol z_k}={1\over 2}(\eta+\ii J\eta).$$
As $\xi,J\xi,\eta,J\eta$ are tangent to $\partial\Omega$, we get on 
$\partial\Omega$~:
$$0=\xi'.(\eta''.\rho)+\eta''.(\xi'.\rho)=
\sum_{1\le j,k\le n}2{\partial^2\rho\over\partial z_j\partial\ol z_k}\,
\xi_j\ol\eta_k+\xi_j{\partial\ol\eta_k\over\partial z_j}
{\partial\rho\over\partial\ol z_k}+\ol\eta_k{\partial\xi_j\over\partial\ol z_k}
{\partial\rho\over\partial z_j}.$$
Since $[\xi,\eta]$ is also tangent to $\partial\Omega$, we have
$\Re\langle[\xi,\eta],\nu\rangle=0$, hence $\langle J[\xi,\eta],\nu\rangle$
is real and
$$\langle[\xi,\eta],J\nu\rangle=-\langle J[\xi,\eta],\nu\rangle=
-{1\over|\nabla\rho|}\big(J[\xi,\eta].\rho\big)=
-{2\over|\nabla\rho|}\Re\big(J[\xi',\eta''].\rho\big)$$
because $J[\xi',\eta']=i[\xi',\eta']$ and its conjugate $J[\xi'',\eta'']$ are 
tangent to $\partial\Omega$. We find now
$$\eqalignno{
J[\xi',\eta'']&=-\ii\sum\xi_j{\partial\ol\eta_k\over\partial z_j}
{\partial\over\partial\ol z_k}+\ol\eta_k{\partial\xi_j\over\partial\ol z_k}
{\partial\over\partial\ol z_j},\cr
\Re\big(J[\xi',\eta''].\rho\big)&=
\Im\sum\xi_j{\partial\ol\eta_k\over\partial z_j}
{\partial\rho\over\partial\ol z_k}+\ol\eta_k
{\partial\xi_j\over\partial\ol z_k}{\partial\rho\over\partial z_j}=-2\Im\sum
{\partial^2\rho\over\partial z_j\partial\ol z_k}\,\xi_j\ol\eta_k,\cr
\langle[\xi,\eta],J\nu\rangle&={4\over|\nabla\rho|}\Im\sum
{\partial^2\rho\over\partial z_j\partial\ol z_k}\,\xi_j\ol\eta_k=
4\Im L_{\partial\Omega}(\xi,\eta).&\square\cr}$$
\endproof

\claim{(7.12) Theorem} An open subset $\Omega\subset\bC^n$ with
$C^2$ boundary is pseudoconvex if and only if the Levi form
$L_{\partial\Omega}$ is semipositive at every point of
$\partial\Omega$.
\endclaim

\proof{Proof} Set $\delta(z)=d(z,\complement\Omega)$, $z\in\ol\Omega$. Then
$\rho=-\delta$ is $C^2$ near $\partial\Omega$ and satisfies (7.9). If 
$\Omega$ is pseudoconvex, the plurisubharmonicity of $-\log(-\rho)$ means that
for all $z\in\Omega$ near $\partial\Omega$ and all $\xi\in\bC^n$ one has
$$\sum_{1\le j,k\le n}\Big({1\over|\rho|}\,{\partial^2\rho\over\partial z_j
\partial\ol z_k}+{1\over\rho^2}\,{\partial\rho\over\partial z_j}\,
{\partial\rho\over\partial\ol z_k}\Big)\xi_j\ol\xi_k\ge 0.$$
Hence $\sum~(\partial^2\rho/\partial z_j\partial\ol z_k)\xi_j\ol\xi_k\ge 0$ if
$\sum~(\partial\rho/\partial z_j)\xi_j=0$, and an easy argument shows
that this is also true at the limit on $\partial\Omega$.

Conversely, if $\Omega$ is not pseudoconvex, Th.~7.2 and 7.7 show
that $-\log\delta$ is not plurisubharmonic in any neighborhood of 
$\partial\Omega$. Hence there exists $\xi\in\bC^n$ such that
$$c=\Big({\partial^2\over\partial t\partial\ol t}\,\log\delta(z+t\xi)
\Big)_{|t=0}>0$$
for some $z$ in the neighborhood of $\partial\Omega$ where $\delta\in C^2$.
By Taylor's formula, we have
$$\log\delta(z+t\xi)=\log\delta(z)+\Re(at+bt^2)+c|t|^2+{\rm o}(|t|^2)$$
with $a,b\in\bC$. Now, choose $z_0\in\partial\Omega$ such that
$\delta(z)=|z-z_0|$ and set
$$h(t)=z+t\xi+e^{at+bt^2}(z_0-z),~~~~t\in\bC.$$
Then we get $h(0)=z_0$ and
$$\eqalign{
\delta(h(t))&\ge\delta(z+t\xi)-\delta(z)\,\big|e^{at+bt^2}\big|\cr
&\ge\delta(z)\,\big|e^{at+bt^2}\big|\,\big(e^{c|t|^2/2}-1\big)
\ge\delta(z)\,c|t|^2/3\cr}$$
when $|t|$ is sufficiently small. Since $\delta(h(0))=\delta(z_0)=0$,
we obtain at $t=0$~:
$${\partial\over\partial t}\,\delta(h(t))=\sum {\partial\delta\over\partial
z_j}(z_0)\,h'_j(0)=0,$$
$${\partial^2\over\partial t\partial\ol t}\,\delta(h(t))=\sum 
{\partial^2\delta\over\partial z_j\partial\ol z_k}(z_0)\,h'_j(0)\ol{h'_k(0)}>0
,$$
hence $h'(0)\in {}^hT_{\partial\Omega,z_0}$ and $L_{\partial\Omega,z_0}
(h'(0))<0$.\qed
\endproof

\claim{(7.13) Definition} The boundary $\partial\Omega$ is said to be
weakly $($resp.\ strongly$)$ pseudoconvex if $L_{\partial\Omega}$ is
semipositive $($resp.\ positive definite$)$ on~$\partial\Omega$. The
boundary is said to be Levi flat if $L_{\partial\Omega}\equiv 0$.
\endclaim

\claim{(7.14) Remark} \rm Lemma 7.11 shows that $\partial\Omega$ is
Levi flat if and only if the subbundle
${}^hT_{\partial\Omega}\subset T_{\partial\Omega}$ is integrable
(i.e.\ stable under the Lie bracket). Assume that $\partial\Omega$ is of
class $\cC^k$, $k\ge 2$. Then ${}^hT_{\partial\Omega}$ is of class
$C^{k-1}$. By Frobenius' theorem, the integrability condition implies
that ${}^hT_{\partial\Omega}$ is the tangent bundle to a $\cC^k$
foliation of $\partial\Omega$ whose leaves have real dimension $2n-2$.
But the leaves themselves must be complex analytic since
${}^hT_{\partial\Omega}$ is a complex vector space (cf.\ Lemma~7.15
below). Therefore  $\partial\Omega$ is Levi flat if and only if it is
foliated by complex analytic hypersurfaces.
\endclaim

\claim{(7.15) Lemma} Let $Y$ be a $C^1$-submanifold of a complex analytic
manifold $X$. If the tangent space $T_{Y,x}$ is a complex subspace of
$T_{X,x}$ at every point $x\in Y$, then $Y$ is complex analytic.
\endclaim

\proof{Proof} Let $x_0\in Y$. Select holomorphic coordinates $(z_1,\ldots,z_n)$
on $X$ centered at $x_0$ such that $T_{Y,x_0}$ is spanned by
$\partial/\partial z_1,\ldots,\partial/\partial z_p$. Then there exists a
neighborhood $U=U'\times U''$ of $x_0$ such that $Y\cap U$ is a graph
$$z''=h(z'),~~~~z'=(z_1,\ldots,z_p)\in U',~~z''=(z_{p+1},\ldots,z_n)$$
with $h\in C^1(U')$ and $dh(0)=0$. The differential of $h$ at $z'$ is
the composite of the projection of $\bC^p\times\{0\}$ on
$T_{Y,(z',h(z'))}$ along $\{0\}\times\bC^{n-p}$ and of the second projection
$\bC^n\to\bC^{n-p}$. Hence $dh(z')$ is $\bC$-linear at every point
and $h$ is holomorphic.\qed
\endproof

\section{8}{Exercises}\begineightpoint
\subsubsection{8.1}{} Let $\Omega\subset\bC^n$ be an open set such that
$$z\in\Omega,~~\lambda\in\bC,~~|\lambda|\le 1\Longrightarrow
\lambda z\in\Omega.$$
Show that $\Omega$ is a union of polydisks of center $0$ (with
arbitrary linear changes of coordinates) and infer that
the space of polynomials $\bC[z_1,\ldots,z_n]$ is dense in $\cO(\Omega)$
for the topology of uniform convergence on compact subsets and in
$\cO(\Omega)\cap C^0(\ol\Omega)$ for the topology of uniform convergence
on~$\ol\Omega$.\\
{\it Hint}\/: consider the Taylor expansion of a function $f\in\cO(\Omega)$
at the origin, writing it as a series of homogeneous polynomials. To
deal with the case of $\ol\Omega$, first apply a dilation to~$f$.

\subsubsection{8.2}{} Let $B\subset\bC^n$ be the unit euclidean ball,
$S=\partial B$ and $f\in\cO(B)\cap C^0(\ol B)$. Our goal is to check
the following Cauchy formula:
$$f(w)={1\over\sigma_{2n-1}}\int_S{f(z)\over(1-\langle w,z\rangle)^n}\,
d\sigma(z).$$
\item{a)} By means of a unitary transformation and Exercise 8.1,
reduce the question to the case when $w=(w_1,0,\ldots,0)$ and $f(z)$ is a
monomial $z^\alpha$.
\smallskip
\item{b)} Show that the integral $\int_B z^\alpha\ol z_1^k\,d\lambda(z)$
vanishes unless $\alpha=(k,0,\ldots,0)$. Compute the value of the
remaining integral by the Fubini theorem, as well as the
integrals $\int_S z^\alpha \ol z_1^k\,d\sigma(z)$.
\smallskip
\item{c)} Prove the formula by a suitable power series expansion.

\subsubsection{8.3}{} A current $T\in\cD'_p(M)$ is said to be {\it normal} if both
$T$ and $dT$ are of order zero, i.e.\ have measure coefficients.
\smallskip
\item{a)} If $T$ is normal and has support contained in a $C^1$ submanifold
$Y\subset M$, show that there exists a normal current $\Theta$ on $Y$
such that $T=j_\star\Theta$, where $j:Y\longrightarrow M$ is the inclusion.\\
{\it Hint}\/: if $x_1=\ldots=x_q=0$ are equations of $Y$ in a coordinate
system $(x_1,\ldots,x_n)$, observe that $x_jT=x_jdT=0$ for $1\le j\le q$ and
infer that $dx_1\wedge\ldots\wedge dx_q$ can be factorized in all terms
of $T$.
\smallskip
\item{b)} What happens if $p>\dim Y$~?
\smallskip
\item{c)} Are a) and b) valid when the normality assumption is dropped~?
 
\subsubsection{8.4}{} Let $T=\sum_{1\le j\le n}T_jd\ol z_j$ be a closed current of
bidegree $(0,1)$ with compact support in $\bC^n$ such that $d''T=0$.
\smallskip
\item{a)} Show that the partial convolution $S=(1/\pi z_1)~\star_1~T_1$
is a solution of the equation $d''S=T$.
\smallskip
\item{b)} Let $K={\rm Supp}\,T$. If $n\ge 2$, show that $S$ has support in 
the compact set $\wt K$ equal to the union of $K$ and of all bounded 
components of $\bC^n\ssm K$.\\
{\it Hint}\/: observe that $S$ is holomorphic on $\bC^n\ssm K$ and
that $S$ vanishes for \hbox{$|z_2|+\ldots+|z_n|$} large.

\subsubsection{8.5}{} Alternative proof of the Dolbeault-Grothendieck lemma.
Let $v=\sum_{|J|=q}\!\!v_Jd\ol z_J$, $q\ge 1$, be a smooth
form of bidegree $(0,q)$ on a polydisk $\Omega=D(0,R)\subset\bC^n$,
such that $d''v=0$, and let $\omega=D(0,r)\compact\omega$. Let $k$ be
the smallest integer such that the monomials $d\ol z_J$ appearing in
$v$ only involve $d\ol z_1$, $\ldots$, $d\ol z_k$. Prove by induction
on $k$ that the equation $d''u=v$ can be solved on $\omega$.\\
{\it Hint}\/: set $v=f\wedge d\ol z_k+g$ where $f$, $g$ only involve
$d\ol z_1$, $\ldots$, $d\ol z_{k-1}$. Then consider $v-d''F$ where
$$F=\sum_{|J|=q-1}F_Jd\ol z_J,~~~~F_J(z)=(\psi(z_k)f_J)\star_k
\Big({1\over \pi z_k}\Big),$$
where $\star_k$ denotes the partial convolution with respect to~$z_k$,
$\psi(z_k)$ is a cut-off function equal to 1 on $D(0,r_k+\varepsilon)$
and $f=\sum_{|J|=q-1}f_Jd\ol z_J$.

\subsubsection{8.6}{} Construct locally bounded non continuous subharmonic 
functions on $\bC$.\\
{\it Hint}\/: consider $e^u$ where $u(z)=\sum_{j\ge 1}2^{-j}\log|z-1/j|$.
 
\subsubsection{8.7}{} Let $\omega$ be an open subset of $\bR^n$, $n\ge 2$,
and $u$ a subharmonic function which is not locally $-\infty$.
\smallskip
\item{a)} For every open set $\omega\compact\Omega$, show that there is a
positive measure $\mu$ with support in $\ol\omega$ and a harmonic function
$h$ on $\omega$ such that $u=N\star\mu+h$ on $\omega$.
\smallskip
\item{b)} Use this representation to prove the following properties:
$u\in L^p_\loc$ for all $p<n/(n-2)$ and $\partial u/\partial x_j
\in L^p_\loc$ for all $p<n/(n-1)$.

\subsubsection{8.8}{} Show that a connected open set $\Omega\subset\bR^n$ is
convex if and only if $\Omega$ has a locally convex exhaustion 
function $\varphi$.\\
{\it Hint}\/: to show the sufficiency, take a path $\gamma:[0,1]\to\Omega$
joining two arbitrary points $a,b\in\Omega$ and consider the restriction 
of $\varphi$ to $[a,\gamma(t_0)]\cap\Omega$ where $t_0$ is the supremum
of all $t$ such that $[a,\gamma(u)]\subset\Omega$ for $u\in[0,t]$.
 
\subsubsection{8.9}{} Let $r_1,r_2\in{}]1,+\infty[$. Consider the compact set
$$K=\{|z_1|\le r_1\,,\,|z_2|\le 1\}\cup\{|z_1|\le 1\,,\,|z_2|\le r_2\}
\subset\bC^2.$$
Show that the holomorphic hull of $K$ in $\bC^2$ is
$$\wh K=\{|z_1|\le r_1\,,\,|z_2|\le r_2\,,\,|z_1|^{1/\log r_1}
|z_2|^{1/\log r_2}\le e\}.$$
{\it Hint}\/: to show that $\wh K$ is contained in this set, consider
all holomorphic monomials $f(z_1,z_2)=z_1^{\alpha_1}z_2^{\alpha_2}$.
To show the converse inclusion, apply the maximum
principle to the domain $|z_1|\le r_1$, $|z_2|\le r_2$ on suitably
chosen Riemann surfaces $z_1^{\alpha_1}z_2^{\alpha_2}=\lambda$.

\subsubsection{8.10}{} Compute the rank of the Levi form of the ellipsoid
$|z_1|^2+|z_3|^4+|z_3|^6<1$ at every point of the boundary.

\subsubsection{8.11}{} Let $X$ be a complex manifold and let $u(z)=\sum_{j\in\bN}
|f_j|^2$, $f_j\in\cO(X)$, be a series converging uniformly on every compact
subset of~$X$. Prove that the limit is real analytic and that the series
remains uniformly convergent by taking derivatives term by term.\\
{\it Hint}\/: since the problem is local, take $X=B(0,r)$, a ball in
$\bC^n$. Let $g_j(z)=\ol{g_j(\ol z)}$ be the conjugate function of
$f_j$ and let $U(z,w)=\sum_{j\in\bN}f_j(z)g_j(w)$ on $X\times X$.
Using the Cauchy-Schwarz inequality, show that this series of holomorphic
functions is uniformly convergent on every compact subset of~$X\times X$.

\subsubsection{8.12}{} Let $\Omega\subset\bC^n$ be a bounded open set with 
$C^2$ boundary.
\smallskip
\item{a)} Let $a\in\partial\Omega$ be a given point.  Let $e_n$ be the
outward normal vector to $T_{\partial\Omega,a}$, $(e_1,\ldots,e_{n-1})$ an
orthonormal basis of ${}^hT_a(\partial\Omega)$ in which the Levi
form is diagonal and $(z_1,\ldots,z_n)$ the associated linear coordinates
centered at~$a$. Show that there is a neighborhood $V$ of $a$ such that
$\partial\Omega\cap V$ is the graph $\Re z_n=-\varphi(z_1,\ldots,z_{n-1},\Im z_n)$
of a function $\varphi$ such that $\varphi(z)=O(|z|^2)$ and
the matrix $\partial^2\varphi/\partial z_j\partial\ol z_k(0)$,
$1\le j,k\le n-1$ is diagonal.
\smallskip
\item{b)} Show that there exist local analytic coordinates 
$w_1=z_1,\ldots,w_{n-1}=z_{n-1}$, $w_n=z_n+\sum c_{jk}z_jz_k$ on a 
neighborhood $V'$ of $a=0$ such that 
$$\Omega\cap V'=V'\cap
\{\Re w_n+\sum_{1\le j\le n}\lambda_j|w_j|^2+{\rm o}(|w|^2)<0\},~~~~
\lambda_j\in\bR$$
and that $\lambda_n$ can be assigned to any given value by a suitable
choice of the coordinates.\\
{\it Hint}\/: Consider the Taylor expansion of order $2$ of the defining
function $\rho(z)=(\Re z_n+\varphi(z))(1+\Re\sum c_jz_j)$ where $c_j\in\bC$
are chosen in a suitable way.
\smallskip
\item{c)} Prove that $\partial\Omega$ is strongly pseudoconvex at $a$
if and only if there is a neighborhood $U$ of $a$ and a biholomorphism $\Phi$
of $U$ onto some open set of $\bC^n$ such that $\Phi(\Omega\cap U)$ is
strongly convex.
\smallskip
\item{d)} Assume that the Levi form of $\partial\Omega$ is not 
semipositive. Show that all holomorphic functions $f\in\cO(\Omega)$ 
extend to some (fixed) neighborhood of $a$.\\
{\it Hint}\/: assume for example $\lambda_1<0$. For $\varepsilon>0$ small,
show that $\Omega$ contains the Hartogs figure
$$\eqalign{
\{\varepsilon/2<|w_1|<\varepsilon\}&\times\{|w_j|<\varepsilon^2\}_{1<j<n}
\times\{|w_n|<\varepsilon^{3/2}\,,\,\Re w_n<\varepsilon^3\}~~\cup\cr
\{|w_1|<\varepsilon\}&\times\{|w_j|<\varepsilon^2\}_{1<j<n}
\times\{|w_n|<\varepsilon^{3/2}\,,\,\Re w_n<-\varepsilon^2\}.\cr}$$
 
\subsubsection{8.13}{} Let $\Omega\subset\bC^n$ be a bounded open set with $C^2$
boundary and $\rho\in C^2(\Omega,\bR)$ such that $\rho<0$ on $\Omega$,
$\rho=0$ and $d\rho\ne 0$ on $\partial\Omega$. Let $f\in 
C^1(\partial\Omega,\bC)$ be a function satisfying the 
{\it tangential Cauchy-Riemann equations}
$$\xi''\cdot f=0,~~~~\forall\xi\in {}^hT_{\partial\Omega},~~~~
\xi''={1\over 2}(\xi+\ii J\xi).$$
\item{a)} Let $f_0$ be a $C^1$ extension of $f$ to $\ol\Omega$. 
Show that $d''f_0\wedge d''\rho=0$ on $\partial\Omega$ and infer that
$v=\bOne_\Omega d''f_0$ is a $d''$-closed current on $\bC^n$.
\smallskip
\item{b)} Show that the solution $u$ of $d''u=v$ provided by
Cor.~3.27 is continuous and that $f$ admits an extension $\wt f\in\cO(\Omega)
\cap C^0(\ol\Omega)$ if $\partial\Omega$ is connected.
 
\subsubsection{8.14}{} Let $\Omega\subset\bC^n$ be a bounded pseudoconvex 
domain with $C^2$ boundary and let $\delta(z)=d(z,\complement\Omega)$
be the euclidean distance to the boundary. 
\smallskip
\item{a)} Use the plurisubharmonicity of $-\log\delta$ to prove the 
following fact: for every $\varepsilon>0$ there is a 
constant $C_\varepsilon>0$ such that
$${-H\delta_z(\xi)\over\delta(z)}+\varepsilon{|d'\delta_z.\xi|^2\over
|\delta(z)|^2}+C_\varepsilon|\xi|^2\ge0$$
for $\xi\in\bC^n$ and $z$ near $\partial\Omega$.
\smallskip
\item{b)} Set $\psi(z)=-\log\delta(z)+K|z|^2$. Show that for $K$ large 
and $\alpha$ small the function 
$$\rho(z)=-\exp\big(-\alpha\psi(z)\big)=-\big(e^{-K|z|^2}\delta(z)
\big)^\alpha$$
is plurisubharmonic.
\smallskip
\item{c)} Prove the existence of a plurisubharmonic exhaustion
function $u:\Omega\to[-1,0[$ of class $C^2$ such that $|u(z)|$ has 
the same order of magnitude as $\delta(z)^\alpha$ when $z$ tends to 
$\partial\Omega$.\\
{\it Hint}\/: consult \cite{Diederich-Fornaess 1976}.
 
\subsubsection{8.15}{} Let $\Omega=\omega+\ii\bR^n$ be a connected tube in
$\bC^n$ of base $\omega$.
\smallskip
\item{a)} Assume first that $n=2$. Let $T\subset\bR^2$ be the triangle 
$x_1\ge 0$, $x_2\ge 0$, $x_1+x_2\le 1$, and assume that the two edges
$[0,1]\times\{0\}$ and $\{0\}\times[0,1]$ are contained in $\omega$. 
Show that every holomorphic function $f\in\cO(\Omega)$
extends to a neighborhood of $T+\ii\bR^2$.\\
{\it Hint}\/: let $\pi:\bC^2\longrightarrow\bR^2$ be the projection on the real
part and $M_\varepsilon$ the intersection of $\pi^{-1}((1+\varepsilon)T)$
with the Riemann surface $z_1+z_2-{\varepsilon\over 2}(z_1^2+z_2^2)=1$ (a
non degenerate affine conic). Show that $M_\varepsilon$ is compact and that
$$\eqalign{
&\pi(\partial M_\varepsilon)\subset([0,1+\varepsilon]\times\{0\})\cup
(\{0\}\times[0,1+\varepsilon])\subset\omega,\cr
&\pi([0,1]\cdot M_\varepsilon)\supset T\cr}$$
for $\varepsilon$ small. Use the Cauchy formula along $\partial M_\varepsilon$ 
(in some parametrization of the conic)
to obtain an extension of $f$ to $[0,1]\cdot M_\varepsilon+\ii\bR^n$.
\smallskip
\item{b)} In general, show that every $f\in\cO(\Omega)$ extends to 
the convex hull $\wh\Omega$.\\
{\it Hint}\/: given $a,b\in\omega$, consider a polygonal line joining $a$ and
$b$ and apply a) inductively to obtain an extension along
$[a,b]+\ii\bR^n$.

\subsubsection{8.16}{} For each integer $\nu\ge 1$, consider the algebraic variety
$$X_\nu=\big\{(z,w,t)\in\bC^3\,;\,wt=p_\nu(z)\big\},~~~~
p_\nu(z)=\prod_{1\le k\le\nu}(z-1/k),$$
and the map $j_\nu:X_\nu\to X_{\nu+1}$ such that
$$j_\nu(z,w,t)=\Big(z,\,w,\,t\Big(z-{1\over\nu+1}\Big)\Big).$$
\item{a)} Show that $X_\nu$ is a Stein manifold, and that $j_\nu$
is an embedding of $X_\nu$ onto an open subset of $X_{\nu+1}$.
\item{b)} Define $X=\lim (X_\nu,j_\nu)$, and let $\pi_\nu:X_\nu\to
\bC^2$ be the projection to the first two coordinates. Since
$\pi_{\nu+1}\circ j_\nu=\pi_\nu$, there exists a holomorphic map
$\pi:X\to\bC^2$, $\pi=\lim\pi_\nu$. Show that
$$\bC^2\ssm\pi(X)=\big\{(z,0)\in\bC^2\,;\,z\ne 1/\nu,~\forall\nu\in\bN,~
\nu\ge 1\big\},$$
and especially, that $(0,0)\notin\pi(X)$.
\item{c)} Consider the compact set
$$K=\pi^{-1}\big(\{(z,w)\in\bC^2\,;\,|z|\le 1,~|w|=1\}\big).$$
By looking at points of the forms $(1/\nu,w,0)$, $|w|=1$, show that
$\pi^{-1}(1/\nu,1/\nu)\in\wh K_{\cO(X)}$. Conclude from this that
$X$ is not holomorphically convex (this example is due to 
\cite{Fornaess~1977}).

\subsubsection{8.17}{} Let $X$ be a complex manifold, and let $\pi:\wt X\to X$ be
a holomorphic unramified covering of~$X$ ($X$ and $\wt X$ are assumed
to be connected).
\item{a)} Let $g$ be a complete riemannian metric on~$X$, and let $\wt d$
be the geodesic distance on $\wt X$ associated to $\wt g=\pi^\star g$ (see
VIII-2.3 for definitions). Show that $\wt g$ is complete and that
$\delta_0(x):=\wt d(x,x_0)$ is a continuous exhaustion function
on~$\wt X$, for any given point $x_0\in\wt X$.
\item{b)} Let $(U_\alpha)$ be a locally finite covering of $X$ by open
balls contained in coordinate open sets, such that all intersections
$U_\alpha\cap U_\beta$ are diffeomorphic to convex open sets (see Lemma
IV-6.9). Let $\theta_\alpha$ be a partition of unity subordinate to the
covering $(U_\alpha)$, and let $\delta_{\varepsilon_\alpha}$ be the
convolution of $\delta_0$ with a regularizing kernel
$\rho_{\varepsilon_\alpha}$ on each piece of $\pi^{-1}(U_\alpha)$ which
is mapped biholomorphically onto $U_\alpha$. Finally, set
$\delta=\sum(\theta_\alpha\circ\pi)\delta_{\varepsilon_\alpha}$.
Show that if $(\varepsilon_\alpha)$ is a collection of sufficiently
small positive numbers, then $\delta$ is a smooth exhaustion function
on~$\wt X$.
\item{c)} Using the fact that $\delta_0$ is $1$-Lipschitz with respect to
$\wt d$, show that derivatives $\partial^{|\nu|}\delta(x)/\partial x^\nu$
of a given order with respect to coordinates in $U_\alpha$ are uniformly
bounded in all components of $\pi^{-1}(U_\alpha)$, at least when $x$ lies
in the compact subset $\Supp\theta_\alpha$. Conclude from this that there
exists a positive hermitian form with continuous coefficients on $X$
such that $H\delta\ge-\pi^\star\gamma$ on $\wt X$.
\item{d)} If $X$ is strongly pseudoconvex, show that $\wt X$ is also
strongly pseudoconvex.\\
{\it Hint}\/: let $\psi$ be a smooth strictly plurisubharmonic exhaustion
function on~$X$. Show that there exists a smooth convex increasing
function $\chi:\bR\to\bR$ such that $\delta+\chi\circ\psi$ is
strictly plurisubharmonic.
\endeightpoint

\chapter{Chapter II}{Coherent Sheaves and Analytic Spaces}

\begineightpoint
The chapter starts with rather general and abstract concepts concerning
sheaves and ringed spaces. Introduced in the decade 1950-1960 by Leray,
Cartan, Serre and Grothendieck, sheaves and ringed spaces have since been
recognized as the adequate tools to handle algebraic varieties and
analytic spaces in a unified framework. We then concentrate ourselves
on the theory of complex analytic functions. The second section is
devoted to a proof of the Weierstrass preparation theorem, which is
nothing but a division algorithm for holomorphic functions. It is used
to derive algebraic properties of the ring $\cO_n$ of germs of
holomorphic functions in~$\bC^n$. Coherent analytic sheaves are then
introduced and the fundamental coherence theorem of Oka is proved. 
Basic properties of analytic sets are investigated in detail: local
parametrization theorem, Hilbert's Nullstellensatz, coherence of the
ideal  sheaf of an analytic set, analyticity of the singular set. The
formalism of complex spaces is then developed and gives a natural
setting for the proof of more global properties (decomposition into
global irreducible components, maximum principle). After a few
definitions concerning cycles, divisors and meromorphic functions, we
investigate the important notion of normal  space and establish the Oka
normalization theorem. Next, the Remmert-Stein extension theorem and
the Remmert proper mapping theorem  on images of analytic sets are
proved by means of semi-continuity results on the rank of morphisms. As
an application, we give a proof of Chow's theorem asserting that every
analytic subset of $\bP^n$  is algebraic. Finally, the concept of
analytic scheme with nilpotent elements is introduced as a
generalization of complex spaces, and we discuss the concepts of
bimeromorphic maps, modifications and blowing-up.
\endeightpoint

\section{1}{Presheaves and Sheaves}
\subsection{1.A}{Main Definitions.}
Sheaves have become a very important tool in analytic or algebraic
geometry as well as in algebraic topology. They are especially useful
when one wants to relate global properties of an object to its local 
properties (the latter being usually easier to establish). We first
introduce the axioms of presheaves and sheaves in full generality and
give some basic examples.

\claim{(1.1) Definition} Let $X$ be a topological space. A
presheaf $\cA$ on $X$ consists of the following data:
\medskip
\item{\rm a)} a collection of non empty sets $\cA(U)$
associated with every open set $U\subset X$,
\smallskip
\item{\rm b)} a collection of maps $\rho_{U,V}:\cA(V)\longrightarrow\cA(U)$
defined whenever $U\subset V$ and satisfying the transitivity property
\smallskip
\item{\rm c)} $\rho_{U,V}\circ\rho_{V,W}=\rho_{U,W}$~~~for~~$U\subset
V\subset W,\qquad\rho_{U,U}=\Id_U$~~~for every~$U$.
\medskip\noindent
The set $\cA(U)$ is called the set of sections of the presheaf $\cA$ over~$U$.
\endclaim

Most often, the presheaf $\cA$ is supposed to carry an additional algebraic
structure. For instance:

\claim{(1.2) Definition} A presheaf $\cA$ is said to be a presheaf of
abelian groups $($resp.\ rings, $R$-modules, algebras$)$ if all sets
$\cA(U)$ are abelian groups $($resp.\ rings, $R$-modules, algebras$)$ and
if the maps $\rho_{U,V}$ are morphisms of these algebraic structures. In
this case, we always assume that $\cA(\emptyset)=\{0\}$.
\endclaim

\claim{(1.3) Example} \rm If we assign to each open set $U\subset X$ the
set $\cC(U)$ of all real valued continuous functions on~$U$ and let
$\rho_{U,V}$ be the obvious restriction morphism $\cC(V)\to\cC(U)$,
then $\cC$ is a presheaf of rings on~$X$. Similarly if $X$ is 
differentiable (resp.\ complex analytic) manifold, there are well
defined presheaves of rings $\cC^k$ of functions of class $\cC^k$
(resp.\ $\cO$) of holomorphic functions) on~$X$. Because of these
examples, the maps $\rho_{U,V}$ in Def.~1.1 are often viewed
intuitively as ``restriction homomorphisms'', although the sets $\cA(U)$
are not necessarily sets of functions defined over~$U$. For the
simplicity of notation we often just write $\rho_{U,V}(f)=
f_{\restriction U}$ whenever $f\in\cA(V)$, $V\supset U$.\qed
\endclaim

For the above presheaves $\cC$, $\cC^k$, $\cO$, the properties of
functions under consideration are purely local. As a consequence, these
presheaves satisfy the following additional {\it gluing axioms}, where
$(U_\alpha)$ and $U=\bigcup U_\alpha$ are arbitrary open subsets of $X\,$:
$$\leqalignno{
&\hbox{\rm If $F_\alpha\in\cA(U_\alpha)$ are such that 
$\rho_{U_\alpha\cap U_\beta,U_\alpha}(F_\alpha)=
 \rho_{U_\alpha\cap U_\beta,U_\beta}(F_\beta)$}&(1.4')\cr
&\hbox{\rm for all $\alpha,\beta$, there exists $F\in\cA(U)$ 
such that $\rho_{U_\alpha,U}(F)=F_\alpha\,$;}\cr}$$
$$\hbox{\rm If $F,G\in\cA(U)$ and $\rho_{U_\alpha,U}(F)=
\rho_{U_\alpha,U}(G)$ for all $\alpha$, then $F=G\,$;}\leqno(1.4'')$$
in other words, local sections over the sets $U_\alpha$ can be glued
together if they coincide in the intersections and the resulting section
on $U$ is uniquely defined. Not all presheaves satisfy $(1.4')$ and
$(1.4'')$:

\claim{(1.5) Example} \rm Let $E$ be an arbitrary set with a
distinguished element~$0$ (e.g.\ an abelian group, a $R$-module,~$\ldots$).
The {\it constant presheaf} $E_X$ on $X$ is defined to be $E_X(U)=E$ for
all~$\emptyset\ne U\subset X$ and $E_X(\emptyset)=\{0\}$, with restriction
maps $\rho_{U,V}=\Id_E$ if $\emptyset\ne U\subset V$ and $\rho_{U,V}=0$
if~$U=\emptyset$. Then axiom $(1.4')$ is not satisfied if $U$ is the union
of two disjoint open sets $U_1$, $U_2$ and $E$ contains a non zero
element.
\endclaim

\claim{(1.6) Definition} A presheaf $\cA$ is said to be a sheaf if it
satisfies the gluing axioms $(1.4')$ and $(1.4'')$.
\endclaim

If $\cA$, $\cB$ are presheaves of abelian groups (or of some other
algebraic structure) on the same space~$X$, a presheaf morphism
$\varphi:\cA\to\cB$ is a collection of morphisms $\varphi_U:\cA(U)\to\cB(U)$
commuting with the restriction morphisms, i.e.\ such that
for each pair $U\subset V$ there is a commutative diagram
$$\eqalign{
&~~\cA(V)\buildo{\displaystyle\varphi_V}\over\longrightarrow\cB(V)\cr
&\rho^\cA_{U,V}\big\downarrow~~\phantom{\longrightarrow}~~\big\downarrow\rho^\cB_{U,V}\cr
&~~\cA(U)\buildo{\displaystyle\varphi_U}\over\longrightarrow\cB(U).\cr}$$
We say that $\cA$ is a subpresheaf of $\cB$ in the case where
$\varphi_U:\cA(U)\subset\cB(U)$ is the inclusion morphism; the commutation
property then means that $\rho^\cB_{U,V}(\cA(V))\subset\cA(U)$ for all $U$,
$V$, and that $\rho^\cA_{U,V}$ coincides with $\rho^\cB_{U,V}$ on~$\cA(V)$.
If $\cA$ is a subpresheaf of a presheaf $\cB$ of abelian groups, there is
a presheaf quotient $\cC=\cB/\cA$ defined by $\cC(U)=\cB(U)/\cA(U)$.
In a similar way, one defines the presheaf kernel (resp.\ presheaf image,
presheaf cokernel) of a presheaf morphism $\varphi:\cA\to\cB$ to be the
presheaves
$$U\mapsto\Ker\varphi_U,\qquad U\mapsto\Im\varphi_U,\qquad
U\mapsto\Coker\varphi_U.$$
The direct sum $\cA\oplus\cB$ of presheaves of abelian groups $\cA$,
$\cB$ is the presheaf $U\mapsto\cA(U)\oplus\cB(U)$, the tensor product
$\cA\otimes\cB$ of presheaves of $R$-modules is
$U\mapsto\cA(U)\otimes_R\cB(U)$, etc $\ldots$

\claim{(1.7) Remark} \rm The reader should take care of the fact that the
presheaf quotient of a sheaf by a subsheaf is not necessarily a sheaf.
To give a specific example, let $X=S^1$ be the unit circle in~$\bR^2$, let
$\cC$ be the sheaf of continuous complex valued functions and $\cZ$ the
subsheaf of integral valued continuous functions (i.e.\ locally constant
functions to $\bZ$). The exponential map
$$\varphi=\exp(2\pi\ii\bu):\cC\longrightarrow\cC^\star$$
is a morphism from $\cC$ to the sheaf $\cC^\star$ of invertible
continuous functions, and the kernel of $\varphi$ is precisely~$\cZ$.
However $\varphi_U$ is surjective for all $U\ne X$ but maps $\cC(X)$
onto the multiplicative subgroup of continuous functions of
$\cC^\star(X)$ of degree~$0$. Therefore the quotient presheaf $\cC/\cZ$
is not isomorphic with $\cC^\star$, although their groups of sections
are the same for all $U\ne X$. Since $\cC^\star$ is a sheaf, we see
that $\cC/\cZ$ does not satisfy property~$(1.4')$.\qed
\endclaim

In order to overcome the difficulty appearing in Example~1.7, it is
necessary to introduce a suitable process by which we can produce a
sheaf from a presheaf. For this, it is convenient to introduce a
slightly modified viewpoint for sheaves.

\claim{(1.8) Definition} If $\cA$ is a presheaf, we define the set
$\wt\cA_x$ of germs of $\cA$ at a point $x\in X$ to be the abstract
inductive limit
$$\wt\cA_x=\lim_{\displaystyle\longrightarrow\atop\scriptstyle U\ni x}
\big(\cA(U),\rho_{U,V}\big).$$
More explicitely, $\wt\cA_x$ is the set of equivalence classes of elements
in the disjoint union $\coprod_{U\ni x}\cA(U)$ taken over all open
neighborhoods $U$ of~$x$, with two elements $F_1\in\cA(U_1)$,
$F_2\in\cA(U_2)$ being equivalent, $F_1\sim F_2$, if and only if there is
a neighborhood $V\subset U_1,\,U_2$ such that $F_{1\restriction V}=
F_{2\restriction V}$, i.e., $\rho_{VU_1}(F_1)=\rho_{VU_2}(F_2)$.
The germ of an element $F\in\cA(U)$ at a point $x\in U$ will be denoted
by~$F_x$.
\endclaim

Let $\cA$ be an arbitrary presheaf. The disjoint union
$\wt\cA=\coprod_{x\in X}\wt\cA_x$ can be equipped with a
natural topology as follows: for every $F\in\cA(U)$, we set
$$\Omega_{F,U}=\big\{F_x~;~x\in U\big\}$$
and choose the $\Omega_{F,U}$ to be a basis of the topology of~$\wt\cA\,$;
note that this family is stable by intersection:
$\Omega_{F,U}\cap\Omega_{G,V}=\Omega_{H,W}$ where $W$ is the (open)
set of points $x\in U\cap V$ at which $F_x=G_x$ and $H=\rho_{W,U}(F)$.
The obvious projection map $\pi:\wt\cA\to X$ which sends $\wt\cA_x$ to
$\{x\}$ is then a local homeomorphism (it is actually a homeomorphism
from $\Omega_{F,U}$ onto~$U$). This leads in a natural way to the
following definition:

\claim{(1.9) Definition} Let $X$ and $\cS$ be topological spaces 
(not necessarily Hausdorff), and let $\pi:\cS\longrightarrow X$ be a mapping such that
\smallskip
\item{\rm a)} $\pi$ maps $\cS$ onto $X~;$
\smallskip
\item{\rm b)} $\pi$ is a local homeomorphism, that is, every point in
$\cS$ has an open neigh\-borhood which is mapped homeomorphically by $\pi$
onto an open subset of~$X$.
\smallskip
\noindent Then $\cS$ is called a sheaf-space on $X$ and $\pi$ is called the 
projection of $\cS$ on~$X$. If $x\in X$, then
$\cS_x=\pi^{-1}(x)$ is called the stalk of $\cS$ at $x$.
\endclaim

If $Y$ is a subset of $X$, we denote by $\Gamma(Y,\cS)$ the set of sections
of $\cS$ on $Y$, i.e.\ the set of continuous functions $F:Y\to\cS$ such that
$\pi\circ F=\Id_Y$. It is clear that the presheaf defined by the collection
of sets $\cS'(U):=\Gamma(U,\cS)$ for all open sets $U\subset X$ together with
the restriction maps $\rho_{U,V}$ satisfies axioms $(1.4')$ and $(1.4'')$,
hence $\cS'$ is a sheaf. The set of germs of $\cS'$ at $x$ is in one-to-one
correspondence with the stalk $\cS_x=\pi^{-1}(x)$, thanks to the local
homeomorphism assumption 1.9~b). This shows that one can associate in a
natural way a sheaf $\cS'$ to every sheaf-space~$\cS$, and that the
sheaf-space $(\cS')^\sim$ can be considered to be identical to the
original sheaf-space $\cS$. Since the assignment $\cS\mapsto\cS'$ from
sheaf-spaces to sheaves is an equivalence of categories, we will usually
omit the prime sign in the notation of $\cS'$ and thus use the same
symbols for a sheaf-space and its associated sheaf of sections; in a
corresponding way, we write $\Gamma(U,\cS)=\cS(U)$ when $U$ is an open set. 

Conversely, given a presheaf $\cA$ on $X$, we have an associated
sheaf-space $\wt\cA$ and an obvious presheaf morphism
$$\cA(U)\longrightarrow\wt\cA'(U)=\Gamma(U,\wt\cA),~~~~
F\longmapsto \wt F=(U\ni x\mapsto F_x).\leqno(1.10)$$
This morphism is clearly injective if and only if $\cA$ satisfies axiom
$(1.4'')$, and it is not difficult to see that $(1.4')$ and $(1.4'')$
together imply surjectivity. Therefore $\cA\to\wt\cA'$ is an isomorphism
if and only if $\cA$ is a sheaf. According to the equivalence of categories
between sheaves and sheaf-spaces mentioned above, we will use from now on
the same symbol $\wt\cA$ for the sheaf-space and its associated
sheaf $\wt\cA'$; one says that $\wt\cA$ is the {\it sheaf
associated with the presheaf}~$\cA$. If $\cA$ itself is a sheaf,
we will again identify $\wt\cA$ and $\cA$, but we will of course keep
the notational difference for a presheaf $\cA$ which is not a sheaf.

\claim{(1.11) Example} \rm The sheaf associated to the constant presheaf
of stalk $E$ over $X$ is the sheaf of locally constant functions~$X\to E$.
This sheaf will be denoted merely by $E_X$ or $E$ if there is no risk
of confusion with the corresponding presheaf. In Example~1.7, we have
$\cZ=\bZ_X$ and the sheaf $(\cC/\bZ_X)^\sim$ associated with the
quotient presheaf $\cC/\bZ_X$ is isomorphic to $\cC^\star$ via the
exponential map.\qed
\endclaim

In the sequel, we usually work in the category of sheaves rather than in
the category of presheaves themselves. For instance, the quotient $\cB/\cA$
of a sheaf $\cB$ by a subsheaf $\cA$ generally refers to the sheaf
associated with the quotient presheaf: its stalks are equal to $\cB_x/\cA_x$,
but a section $G$ of $\cB/\cA$ over an open set $U$ need not necessarily come
from a global section of $\cB(U)\,$; what can be only said is that there is a
covering $(U_\alpha)$ of $U$ and local sections $F_\alpha\in\cB(U_\alpha)$
representing $G_{\restriction U_\alpha}$ such that
$(F_\beta-F_\alpha)_{\restriction U_\alpha\cap U_\beta}$ belongs to
$\cA(U_\alpha\cap U_\beta)$. A sheaf morphism $\varphi:\cA\to\cB$ is said
to be injective (resp.\ surjective) if the germ morphism
$\varphi_x:\cA_x\to\cB_x$ is injective (resp.\ surjective) for every
$x\in X$. Let us note again that a surjective sheaf morphism $\varphi$
does not necessarily give rise to surjective morphisms
$\varphi_U:\cA(U)\to\cB(U)$.

\subsection{1.B}{Direct and Inverse Images of Sheaves}
Let $X$, $Y$ be topological spaces and let $f:X\to Y$ be a continuous map.
If $\cA$ is a presheaf on $X$, the {\it direct image} $f_\star\cA$ is the
presheaf on $Y$ defined by
$$f_\star\cA(U)=\cA\big(f^{-1}(U)\big)\leqno(1.12)$$
for all open sets $U\subset Y$. When $\cA$ is a sheaf, it is clear that
$f_\star\cA$ also satisfies axioms $(1.4')$ and $(1.4'')$, thus
$f_\star\cA$ is a sheaf. Its stalks are given by
$$(f_\star\cA)_y=\lim_{\displaystyle\longrightarrow\atop\scriptstyle V\ni y}
\cA\big(f^{-1}(V)\big)\leqno(1.13)$$
where $V$ runs over all open neighborhoods of $y\in Y$.

Now, let $\cB$ be a sheaf on~$Y$, viewed as a sheaf-space with projection
map $\pi:\cB\to Y$. We define the {\it inverse image} $f^{-1}\cB$ by
$$f^{-1}\cB=\cB\times_Y X=\big\{(s,x)\in\cB\times X\,;~\pi(s)=f(x)\big\}
\leqno(1.14)$$
with the topology induced by the product topology on $\cB\times X$. It
is then easy to see that the projection $\pi'={\rm pr}_2:f^{-1}\cB\to X$
is a local homeomorphism, therefore $f^{-1}\cB$ is a sheaf on~$X$.
By construction, the stalks of $f^{-1}\cB$ are
$$(f^{-1}\cB)_x=\cB_{f(x)},\leqno(1.15)$$
and the sections $\sigma\in f^{-1}\cB(U)$ can be considered as
continuous mappings \hbox{$s:U\to\cB$} such that $\pi\circ\sigma=f$.
In particular, any section $s\in\cB(V)$ on an open set $V\subset Y$ has
a {\it pull-back}
$$f^\star s=s\circ f~~\in~~f^{-1}\cB\big(f^{-1}(V)\big).\leqno(1.16)$$
There are always natural sheaf morphisms
$$f^{-1}f_\star\cA\longrightarrow\cA,~~~~\cB\longrightarrow f_\star f^{-1}\cB\leqno(1.17)$$
defined as follows. A germ in $(f^{-1}f_\star\cA)_x=(f_\star\cA)_{f(x)}$
is defined by a local section $s\in (f_\star\cA)(V)=\cA(f^{-1}(V))$
for some neighborhood $V$ of $f(x)\,$; this section can be mapped to
the germ $s_x\in\cA_x$. In the opposite direction, the pull-back
$f^\star s$ of a section $s\in\cB(V)$ can be seen by (1.16) as a section
of $f_\star f^{-1}\cB(V)$. It is not difficult to see that these natural
morphisms are not isomorphisms in general. For instance, if $f$ is
a finite covering map with $q$ sheets and if we take $\cA=E_X$, $\cB=E_Y$
to be constant sheaves, then $f_\star E_X\simeq E^q_Y$ and $f^{-1}E_Y=E_X$,
thus $f^{-1}f_\star E_X\simeq E^q_X$ and $f_\star f^{-1}E_Y\simeq E_Y^q$.

\subsection{1.C}{Ringed Spaces}
Many natural geometric structures considered in analytic or algebraic
geometry can be described in a convenient way as topological spaces
equipped with a suitable ``structure sheaf'' which, most often,
is a sheaf of commutative rings. For instance, a lot of properties of
$\cC^k$ differentiable (resp.\ real analytic, complex analytic) manifolds
can be described in terms of their sheaf of rings $\cC^k_X$ of
differentiable functions (resp.\ $\cC^\omega_X$ of real analytic
functions, $\cO_X$ of holomorphic functions). We first recall a few
standard definitions concerning rings, referring to textbooks on
algebra for more details (see e.g.\ \cite{Lang 1965}).

\claim{(1.18) Some definitions and conventions about rings}
All our rings $R$ are supposed implicitly to have a unit element $1_R$
$($if $R=\{0\}$, we agree that \hbox{$1_R=0_R\,)$}, and a ring morphism
$R\to R'$ is supposed to map $1_R$ to $1_{R'}$. In the subsequent
definitions, we assume that all rings under consideration are commutative.
\smallskip
\item{\rm a)} An ideal $I\subset R$ is said to be prime if $xy\in I$ implies
$x\in I$ or $y\in I$, i.e., if the quotient ring $R/I$ is entire.
\smallskip
\item{\rm b)} An ideal $I\subset R$ is said to be maximal if $I\ne R$ and
there are no ideals $J$ such that $I\subsetneq J\subsetneq R$
$($equiva\-lently, if the quotient ring $R/I$ is a field$)$.
\smallskip
\item{\rm c)} The ring $R$ is said to be a local ring if $R$ has a unique
maximal ideal $\gm$ $($equiva\-lently, if $R$ has an ideal $\gm$ such that
all elements of $R\ssm\gm$ are invertible$)$. Its residual field is
defined to be the quotient field~$R/\gm$.
\smallskip
\item{\rm d)} The ring $R$ is said to be Noetherian if every ideal
$I\subset R$ is finitely gene\-rated $($equiva\-lently, if every increasing
sequence of ideals $I_1\subset I_2\subset\ldots$ is~stationary$)$.
\smallskip
\item{\rm e)} The radical $\sqrt{I}$ of an ideal $I$ is the set of all
elements $x\in R$ such that some power $x^m$, $m\in\bN^\star$, lies in
in~$I$. Then $\sqrt{I}$ is again an ideal of~$R$.
\smallskip
\item{\rm f)} The nilradical $N(R)=\sqrt{\{0\}}$ is the ideal of
nilpotent elements of~$R$. The ring $R$ is said to be reduced if
$N(R)=\{0\}$. Otherwise, its reduction is defined to be the
reduced ring~$R/N(R)$.
\vskip0pt
\endclaim

We now introduce the general notion of a ringed space. 

\claim{(1.19) Definition} A ringed space is a pair $(X,\cR_X)$ consisting 
of a topolo\-gical space $X$ and of a sheaf of rings $\cR_X$ on $X$, called
the structure sheaf. A~morphism
$$F:(X,\cR_X)\to(Y,\cR_Y)$$
of ringed spaces is a pair $(f,F^\star)$ where $f:X\to Y$ is a
continuous map and
$$F^\star~:~~f^{-1}\cR_Y\to\cR_X,~~~~F^\star_x~:~~\cR_{Y,f(x)}
\to\cR_{X,x}$$
a homomorphism of sheaves of rings on $X$, called the comorphism of $F$.
\endclaim

If $F:(X,\cR_X)\to(Y,\cR_Y)$ and $G:(Y,\cR_Y)\to(Z,\cR_Z)$ are
morphisms of ringed spaces, the composite $G\circ F$ is the
pair consisting of the map $g\circ f:X\to Z$ and of the comorphism
$(G\circ F)^\star=F^\star\circ f^{-1}G^\star\,$:
$$\cmalign{
\hfill&F^\star\circ f^{-1}G^\star~:~~&f^{-1}g^{-1}\cR_Z~\,&\kern-1.9mm
\buildo f^{-1}G^\star\over\llra f^{-1}\cR_Y
&\kern-1.9mm\buildo F^\star\over{\relbar\mkern-4mu\longrightarrow}\cR_X,\cr
&F^\star_x\circ G^\star_{f(x)}~:~~~&\cR_{Z,g\circ f(x)}~\,&{\relbar\mkern-4mu
\relbar\mkern-4mu\longrightarrow}~\cR_{Y,f(x)}~&{\relbar\mkern-4mu\longrightarrow}~\cR_{X,x}.\cr}
\leqno(1.20)$$
We say of course that $F$ is an isomorphism of ringed spaces
if there exists $G$ such that $G\circ F=\Id_X$ and $F\circ G=\Id_Y$.

If $(X,\cR_X)$ is a ringed space, the nilradical of $\cR_X$  defines
an ideal subsheaf $\cN_X$ of $\cR_X$, and the identity map
$\Id_X:X\to X$ together with the ring homomorphism $\cR_X\to\cR_X/\cN_X$
defines a ringed space morphism
$$(X,\cR_X/\cN_X)\to (X,\cR_X)\leqno(1.21)$$
called the {\it reduction morphism}. Quite often, the letter $X$ by
itself is used to denote the ringed space $(X,\cR_X)\,$; we then denote
by $X_\red=(X,\cR_X/\cN_X)$ its reduction. The ringed space $X$ is said
to be {\it reduced} if $\cN_X=0$, in which case the reduction morphism
$X_\red\to X$ is an isomorphism. In all examples considered later on in
this book, the structure sheaf $\cR_X$ will be a sheaf of {\it local
rings} over some field~$k$. The relevant definition is as follows.

\claim{(1.22) Definition} \smallskip
\item{\rm a)} A local ringed space is a ringed space
$(X,\cR_X)$ such that all stalks $\cR_{X,x}$ are local rings. The maximal
ideal of~$\cR_{X,x}$ will be denoted by $\gm_{X,x}$. A morphism
$F=(f,F^\star):(X,\cR_X)\to(Y,\cR_Y)$ of local ringed spaces is a
morphism of ringed spaces such that $F^\star_x(\gm_{Y,f(x)})
\subset\gm_{X,x}$ at any point $x\in X$ $($i.e., $F^\star_x$ is a
``local'' homomorphism of rings$)$.
\smallskip
\item{\rm b)} A local ringed space over a field $k$ is a local ringed
space $(X,\cR_X)$ such that all rings $\cR_{X,x}$ are local
$k$-algebras with residual field $\cR_{X,x}/\gm_{X,x}\simeq k$.
A morphism $F$ between such spaces is supposed to have its comorphism
defined by local $k$-homomorphisms $F^\star_x:\cR_{Y,f(x)}\to\cR_{X,x}$.
\vskip0pt
\endclaim

If $(X,\cR_X)$ is a local ringed space over~$k$, we can associate to each
section $s\in\cR_X(U)$ a function
$$\ol s:U\to k,\qquad x\mapsto\ol s(x)\in k=\cR_{X,x}/\gm_{X,x},$$
and we get a sheaf morphism $\cR_X\to\ol\cR_X$ onto a subsheaf of rings
$\ol\cR_X$ of the sheaf of functions from $X$ to $k$. We clearly
have a factorization
$$\cR_X\to\cR_X/\cN_X\to\ol\cR_X,$$
and thus a corresponding factorization of ringed space morphisms
(with $\Id_X$ as the underlying set theoretic map)
$$X_\stred\to X_\red\to X$$
where $X_\stred=(X,\ol R_X)$ is called the {\it strong reduction} of
$(X,\cR_X)$. It is easy to see that $X_\stred$ is actually a reduced
local ringed space over~$k$. We say that $X$ is strongly reduced if
$\cR_X\to\ol\cR_X$ is an isomorphism, that is, if $\cR_X$ can be
identified with a subsheaf of the sheaf of functions $X\to k$
(in our applications to the theory of algebraic or analytic schemes,
the concepts of reduction and strong reduction will actually be the
same$\,$; in general, these notions differ, see Exercise ??.??).
It is important to observe that reduction (resp.\ strong reduction) is
a fonctorial process:\\
if $F=(f,F^\star):(X,\cR_X)\to(Y,\cR_Y)$ is a morphism of ringed spaces
(resp.\ of local ringed spaces over~$k$), there are natural reductions
$$\eqalign{
F_\red&=(f,F^\star_\red):X_\red\to Y_\red,\quad
F^\star_\red:\cR_{Y,f(x)}/\cN_{Y,f(x)}\to\cR_{X,x}/\cN_{X,x},\cr
F_\stred&=(f,f^\star):X_\stred\to Y_\stred,\quad
f^\star:\ol\cR_{Y,f(x)}\to\ol\cR_{X,x},\quad\ol s\mapsto
\ol s\circ f\cr}$$
where $f^\star$ is the usual pull-back comorphism associated with~$f$.
Therefore, if $(X,\cR_X)$ and $(Y,\cR_Y)$ are strongly reduced, the
morphism $F$ is completely determined by the underlying set-theoretic
map~$f$. Our first basic examples of (strongly reduced) ringed spaces are
the various types of mani\-folds already defined in Chapter~I. The language
of ringed spaces provides an equivalent but more elegant and more
intrinsic definition.

\claim{(1.23) Definition} Let $X$ be a Hausdorff separable topological
space. One can define the category of $\cC^k$, $k\in\bN\cup\{\infty,\omega\}$,
differentiable manifolds $($resp.\ complex analytic manifolds$)$ to be the
category of reduced local ringed spaces $(X,\cR_X)$ over~$\bR$
$($resp.\ over~$\bC)$, such that every point $x\in X$ has a neighborhood
$U$ on which the restriction $(U,\cR_{X\restriction U})$ is isomorphic to
a ringed space $(\Omega,\cC^k_\Omega)$ where $\Omega\subset\bR^n$ is
an open set and $\cC^k_\Omega$ is the sheaf of $\cC^k$ differentiable
functions $($resp.\ $(\Omega,\cO_\Omega)$, where $\Omega\subset\bC^n$
is an open subset, and $\cO_\Omega$ is the sheaf of holomorphic functions
on~$\Omega)$.
\endclaim

We say that the ringed spaces $(\Omega,\cC^k_\Omega)$ and
$(\Omega,\cO_\Omega)$ are the {\it models} of the category of
differentiable (resp.\ complex analytic) manifolds, and that a general
object $(X,\cR_X)$ in the category is {\it locally isomorphic} to one of
the given model spaces. It is easy to see that the corresponding ringed
spaces morphisms are nothing but the usual concepts of differentiable and
holomorphic maps.

\subsection{1.D}{Algebraic Varieties over a Field}
As a second illustration of the notion of ringed space, we present here a
brief introduction to the formalism of algebraic varieties, referring to
\cite{Hartshorne 1977} or \cite{EGA 1967} for a much more detailed 
exposition. Our hope is that the reader who already has some
background of analytic or algebraic geometry will find some hints of
the strong interconnections between both theories. Beginners are
invited to skip this section and proceed directly to the theory of
complex analytic sheaves in~\S$,$2.  All rings or algebras occurring
in this section are supposed to be commutative rings with unit.

\subsubsection{1.D.1}{Affine Algebraic Sets.}
Let $k$ be an algebraically closed field of any characteristic. An {\it
affine algebraic set} is a subset $X\subset k^N$ of the affine
space $k^N$ defined by an arbitrary collection $S\subset k[T_1,\ldots,T_N]$ of
polynomials, that is,
$$X=V(S)=\big\{(z_1,\ldots,z_N)\in k^N\,;\,P(z_1,\ldots,z_N)=0,\,
\forall P\in S\big\}.$$
Of course, if $J\subset k[T_1,\ldots,T_N]$ is the ideal generated by $S$, then
$V(S)=V(J)$. As $k[T_1,\ldots,T_N]$ is Noetherian, $J$ is generated by
finitely many elements $(P_1,\ldots,P_m)$, thus $X=V(\{P_1,\ldots,P_m\})$ is always
defined by finitely many equations. Conversely, for any subset $Y\subset
k^N$, we consider the ideal $I(Y)$ of $k[T_1,\ldots,T_N]$, defined by
$$I(Y)=\big\{P\in k[T_1,\ldots,T_N]\,;\,P(z)=0,~\forall z\in Y\big\}.$$
Of course, if $Y\subset k^N$ is an algebraic set, we have $V(I(Y))=Y$.
In the opposite direction, we have the following fundamental result.

\claim{(1.24) Hilbert's Nullstellensatz {\rm (see \cite{Lang 1965})}}
If $J\subset k[T_1,\ldots,T_N]$ is an ideal, then $I(V(J))=\sqrt{J}$.
\endclaim

If $X=V(J)\subset k^N$ is an affine algebraic set, we define the (reduced)
ring $\cO(X)$ of algebraic functions on $X$ to be the set of all
functions $X\to k$ which are restrictions of polynomials, i.e.,
$$\cO(X)=k[T_1,\ldots,T_N]/I(X)=k[T_1,\ldots,T_N]/\sqrt{J}.\leqno(1.25)$$
This is clearly a reduced $k$-algebra. An (algebraic) morphism of affine
algebraic sets $X=V(J)\subset k^N$, $Y=V(J')\subset k^{N'}$ is a map
$f:Y\to X$ which is the restriction of a polynomial map $k^{N'}to k^N$.
We then get a $k$-algebra homomomorphism
$$f^\star:\cO(X)\to\cO(Y),\qquad s\mapsto s\circ f,$$
called the {\it comorphism} of~$f$. In this way, we have defined
a contravariant fonctor
$$X\mapsto\cO(X),\qquad f\mapsto f^\star\leqno(1.26)$$
from the category of affine algebraic sets to the category of 
finitely generated reduced $k$-algebras.

We are going to show the existence of a natural fonctor going in the
opposite direction. In fact, let us start with an arbitrary finitely
generated algebra~$A$ (not necessarily reduced at this moment).
For any choice of gene\-rators $(g_1,\ldots,g_N)$ of $A$ we get a surjective
morphism of the polynomial ring $k[T_1,\ldots,T_N]$ onto~$A$,
$$k[T_1,\ldots,T_N]\to A,\qquad T_j\mapsto g_j,$$
and thus $A\simeq k[T_1,\ldots,T_N]/J$ with the ideal $J$ being the kernel of
this morphism. It is well-known that every maximal ideal $\gm$ of $A$ has
codimension~$1$ in $A$ (see \cite{Lang 1965}), so that $\gm$ gives rise to a
$k$-algebra homomorphism $A\to A/\gm=k$. We thus get a bijection
$$\Hom_\alg(A,k)\to\Spm(A),\qquad u\mapsto\Ker u$$
between the set of $k$-algebra homomorphisms and the set $\Spm(A)$ of
maximal ideals of~$A$. In fact, if $A=k[T_1,\ldots,T_N]/J$, an element
$\varphi\in\Hom_\alg(A,k)$ is completely determined by the values
$z_j=\varphi(T_j\mod J)$, and the corresponding algebra homomorphism
$k[T_1,\ldots,T_N]\to k$, $P\mapsto P(z_1,\ldots,z_N)$ can be factorized mod $J$
if and only if $z=(z_1,\ldots,z_N)\in k^N$ satisfies
the equations
$$P(z_1,\ldots,z_N)=0,\qquad\forall P\in J.$$
We infer from this that 
$$\Spm(A)\simeq V(J)=\big\{(z_1,\ldots,z_N)\in k^N\,;\,P(z_1,\ldots,z_N)=0,\,
\forall P\in J\big\}$$
can be identified with the {\it affine algebraic set} $V(J)\subset k^N$.
If we are given an algebra homomorphism $\Phi:A\to B$ of 
finitely generated $k$-algebras we get a corresponding map
$\Spm(\Phi):\Spm(B)\to\Spm(A)$ described either as
$$\eqalign{
&\Spm(B)\to\Spm(A),\quad\gm\mapsto\Phi^{-1}(\gm)\quad\hbox{or}\cr
&\Hom_\alg(B,k)\to\Hom_\alg(A,k),\quad v\mapsto v\circ\Phi.\cr}$$
If $B=k[T'_1,\ldots,T'_{N'}]/J'$ and $\Spm(B)=V(J')\subset k^{N'}$, it is
easy to see that $\Spm(\Phi):\Spm(B)\to\Spm(A)$ is the restriction of
the polynomial map
$$f:k^{N'}\to k^N,\qquad w\mapsto f(w)=(P_1(w),\ldots,P_N(w)),$$
where $P_j\in k[T'_1,\ldots,T'_{N'}]$ are polynomials such that
$P_j=\Phi(T_j)\mod J'$ in~$B$. We have in this way defined a contravariant
fonctor
$$A\mapsto\Spm(A),\qquad \Phi\mapsto\Spm(\Phi)\leqno(1.27)$$
from the category of finitely generated $k$-algebras to the category
of affine algebraic sets.

Since $A=k[T_1,\ldots,T_N]/J$ and its reduction $A/N(A)=k[T_1,\ldots,T_N]/\sqrt{J}$
give rise to the same algebraic set
$$V(J)=\Spm(A)=\Spm(A/N(A))=V(\sqrt{J}),$$
we see that the category of affine algebraic sets is actually equivalent
to the subcategory of {\it reduced} finitely generated $k$-algebras.

\claim{(1.28) Example} \rm The simplest example of an affine algebraic
set is the affine space
$$k^N=\Spm(k[T_1,\ldots,T_N]),$$
in particular $\Spm(k)=k^0$ is just one point. We agree that
$\Spm(\{0\})=\emptyset$ (observe that $V(J)=\emptyset$ when $J$
is the unit ideal in $k[T_1,\ldots,T_N])$. 
\endclaim

\subsubsection{1.D.2}{Zariski Topology and Affine Algebraic Schemes.}
Let $A$ be a finitely generated algebra and $X=\Spm(A)$. To each
ideal $\ga\subset A$ we associate the zero variety $V(\ga)\subset X$ which
consists of all elements $\gm\in X=\Spm(A)$ such that $\gm\supset \ga\,$;
if
$$A\simeq k[T_1,\ldots,T_N]/J\quad\hbox{and}\quad X\simeq V(J)\subset k^N,$$
then $V(\ga)$ can be identified with the zero variety $V(J_\ga)\subset X$ of
the inverse image $J_\ga$ of $\ga$ in $k[T_1,\ldots,T_N]$. For any family
$(\ga_\alpha)$ of ideals in $A$ we have
$$V(\sum\ga_\alpha)=\bigcap V(\ga_\alpha),\qquad
V(\ga_1)\cup V(\ga_2)=V(\ga_1\ga_2),$$
hence there exists a unique topology on $X$ such that the closed sets consist
precisely of all algebraic subsets $(V(\ga))_{\ga\subset A}$ of~$X$. This
topology is called the Zariski topology. The Zariski topology is almost
never Hausdorff (for example, if $X=k$ is the affine line, the open sets are
$\emptyset$ and complements of finite sets, thus any two nonempty open sets
have nonempty intersection). However, $X$ is a {\it Noetherian space},
that is, a topological space in which every decreasing sequence of
closed sets is stationary; an equivalent definition is to require that
every open set is quasi-compact (from any open covering of an open set,
one can extract a finite covering). 

We now come to the concept of affine open subsets. For $s\in A$,
the open set $D(s)=X\ssm V(s)$ can be given the structure of an affine
algebraic variety. In fact, if $A=k[T_1,\ldots,T_N]/J$ and $s$ is represented
by a polynomial in $k[T_1,\ldots,T_N]$, the localized ring
$A[1/s]$ can be written as $A[1/s]=k[T_1,\ldots,T_N,T_{N+1}]/J_s$
where $J_s=J[T_{N+1}]+(sT_{N+1}-1)$, thus
$$V(J_s)=\{(z,w)\in V(J)\times k\,;\, s(z)\,w=1\}\simeq V(I)\ssm s^{-1}(0)$$
and $D(s)$ can be identified with $\Spm(A[1/s])$. We have
$D(s_1)\cap D(s_2)=D(s_1s_2)$, and the sets $(D(s))_{s\in A}$
are easily seen to be a basis of the Zariski topology on~$X$. The open sets
$D(s)$ are called {\it affine open sets}. Since the open sets $D(s)$
containing a given point $x\in X$ form a basis of neighborhoods,
one can define a sheaf space $\cO_X$ such that the ring of germs
$\cO_{X,x}$ is the inductive limit
$$\cO_{X,x}=\lim_{\displaystyle\longrightarrow\atop\scriptstyle D(s)\ni x}A[1/s]=
\{\hbox{fractions}~p/q\,;\,p,q\in A,\,q(x)\ne 0\}.$$
This is a local ring with maximal ideal
$$\gm_{X,x}=\{p/q\,;\,p,q\in A,\,p(x)=0,\,q(x)\ne 0\},$$
and residual field $\cO_{X,x}/\gm_{X,x}=k$.
In this way, we get a ringed space $(X,\cO_X)$ over~$k$.
It is easy to see that $\Gamma(X,\cO_X)$ coincides with the
finitely generated $k$-algebra~$A$. In fact, from the definition
of~$\cO_X$, a global section is obtained
by gluing together local sections $p_j/s_j$ on affine open sets $D(s_j)$
with \hbox{$\bigcup D(s_j)=X$}, $1\le j\le m$. This means that the ideal
$\ga=(s_1,\ldots,s_m)\subset A$ has an empty zero variety $V(\ga)$, thus $\ga=A$
and there are elements $u_j\in A$ with $\sum u_js_j=1$. The compatibility
condition $p_j/s_j=p_k/s_k$ implies that these elements are induced by
$$\sum u_jp_j/\sum u_js_j=\sum u_jp_j\in A,$$
as desired. More generally, since the open sets $D(s)$ are affine, we get
$$\Gamma(D(s),\cO_X)=A[1/s].$$
It is easy to see that the ringed space $(X,\cO_X)$ is reduced if and only
if $A$ itself is reduced; in this case, $X$ is even strongly reduced
as Hilbert's Nullstellensatz shows. Otherwise, the reduction $X_\red$ can
obtained from the reduced algebra $A_\red=A/N(A)$.

Ringed spaces $(X,\cO_X)$ as above are called {\it affine algebraic
schemes} over~$k$ (although substantially different from the usual
definition, our definition can be shown to be equivalent in this
special situation; compare with \cite{Hartshorne 1977}); see also
Exercise ??.??). The category of affine algebraic schemes is equivalent to
the category of finitely generated $k$-algebras (with the arrows reversed).

\subsubsection{1.D.3}{Algebraic Schemes.}
Algebraic schemes over $k$ are defined to be ringed spaces over
$k$ which are locally isomorphic to affine algebraic schemes, modulo
an ad hoc separation condition.

\claim{(1.29) Definition} An algebraic scheme over $k$ is a local
ringed space $(X,\cO_X)$ over $k$ such that
\smallskip
\item{\rm a)} $X$ has a finite covering by open sets $U_\alpha$ such that
$(U_\alpha,\cO_{X\restriction U_\alpha})$ is isomorphic as a ringed
space to an affine algebraic scheme $(\Spm(A_\alpha),\cO_{\Spm(A_\alpha)})$.
\smallskip
\item{\rm b)} $X$ satisfies the algebraic separation axiom, namely
the diagonal $\Delta_X$ of $X\times X$ is closed for the Zariski topology.
\medskip\noindent
A morphism of algebraic schemes is just a morphism of the underlying
local ringed spaces. An $($abstract$)$ algebraic variety is the same
as a reduced algebraic scheme.
\vskip0pt
\endclaim

In the above definition, some words of explanation are needed for b),
since the product $X\times Y$ of algebraic schemes over $k$ {\it is not}
the ringed space theoretic product, i.e., the product topological space
equipped with the structure sheaf $\pr_1^\star\cO_X\otimes_k\pr_2^\star\cO_Y$.
Instead, we define the product of two affine algebraic schemes $X=\Spm(A)$
and $Y=\Spm(B)$ to be $X\times Y=\Spm(A\otimes_k B)$, equipped with the
Zariski topology and the structural sheaf associated with~$A\otimes_k B$.
Notice that the Zariski topology on $X\times Y$ {\it is not the
product topology} of the Zariski topologies on $X$,~$Y$, as the example
$k^2=k\times k$ shows; also, the rational function $1/(1-z_1-z_2)\in
\cO_{k^2,(0,0)}$ is not in $\cO_{k,0}\otimes_k\cO_{k,0}$. In general,
if $X$, $Y$ are written as $X=\bigcup U_\alpha$ and $Y=\bigcup V_\beta$ with
affine open sets $U_\alpha$, $V_\beta$, we define $X\times Y$
to be the union of all open affine charts $U_\alpha\times V_\beta$
with their associated structure sheaves of affine algebraic varieties,
the open sets of $X\times Y$ being all unions of open sets in the various
charts $U_\alpha\times V_\beta$. The separation axiom b) is introduced for
the sake of excluding pathological examples such as an affine line
$k\amalg\{0'\}$ with the origin changed into a double point.

\subsubsection{1.D.4}{Subschemes.}
If $(X,\cO_X)$ is an affine algebraic scheme and $A=\Gamma(X,\cO_X)$ is the
associated algebra, we say that $(Y,\cO_Y)$ is a {\it subscheme} of
$(X,\cO_X)$ if there is an ideal $\ga$ of $A$ such that $Y\hookrightarrow X$
is the morphism defined by the algebra morphism $A\to A/\ga$ as its
comorphism. As $\Spm(A/\ga)\to\Spm(A)$ has for image the set $V(\ga)$ of
maximal ideals $\gm$ of $A$ containing $\ga$, we see that $Y=V(\ga)$ as a
set; let us introduce the ideal subsheaf $\cJ=\ga\cO_X\subset\cO_X$.
Since the structural sheaf $\cO_Y$ is obtained by taken localizations
$A/\ga[1/s]$, it is easy to see that $\cO_Y$ coincides with the quotient
sheaf $\cO_X/\cJ$ restricted to~$Y$. Since $\ga$ has finitely many
generators, the ideal sheaf $\cJ$ is locally finitely generated
(see \S~2 below). This leads to the following definition.

\claim{(1.30) Definition} If $(X,\cO_X)$ is an algebraic scheme, a
$($closed$)$ subscheme is an algebraic scheme $(Y,\cO_Y)$ such that
$Y$ is a Zariski closed subset of~$X$, and there is a locally finitely
generated ideal subsheaf $\cJ\subset\cO_X$ such that $Y=V(\cJ)$ and
$\cO_Y=(\cO_X/\cJ)_{\restriction Y}$.
\endclaim

If $(Y,\cO_Y)$, $(Z,\cO_Z)$ are subschemes of $(X,\cO_X)$ defined
by ideal subsheaves $\cJ,\,\cJ'\subset\cO_X$, there are
corresponding subschemes $Y\cap Z$ and $Y\cup Z$ defined as ringed spaces
$$(Y\cap Z,\cO_X/(\cJ+\cJ')),\qquad(Y\cup Z,\cO_X/\cJ\cJ').$$

\subsubsection{1.D.5}{Projective Algebraic Varieties.}
A very important subcategory of the category of algebraic varieties is
provided by {\it projective algebraic varieties}. Let $\bP^N_k$ be the
projective $N$-space, that is, the set $k^{N+1}\ssm\{0\}/k^\star$ of
equivalence classes of $(N+1)$-tuples $(z_0,\ldots,z_N)\in k^{N+1}\ssm\{0\}$
under the equivalence relation given by $(z_0,\ldots,z_N)\sim\lambda
(z_0,\ldots,z_N)$, $\lambda\in k^\star$. The corresponding element of
$\bP^N_k$ will be denoted $[z_0:z_1:\ldots:z_N]$. It is clear that
$\bP_N^k$ can be covered by the $(N+1)$ affine charts $U_\alpha$,
$0\le\alpha\le N$, such that
$$U_\alpha=\big\{[z_0:z_1:\ldots:z_N]\in\bP^N_k\,\;\,z_\alpha\ne 0\big\}.$$
The set $U_\alpha$ can be identified with the affine $N$-space $k^N$ by
the map
$$U_\alpha\to k^N,\qquad[z_0:z_1:\ldots:z_N]\mapsto
\Big({z_0\over z_\alpha},\,{z_1\over z_\alpha},\ldots,
{z_{\alpha-1}\over z_\alpha},\,{z_{\alpha+1}\over z_\alpha},\ldots,
{z_N\over z_\alpha}\Big).$$
With this identification, $\cO(U_\alpha)$ is the algebra of
homogeneous rational functions of degree $0$ in $z_0,\ldots,z_N$ which have just
a power of $z_\alpha$ in their denominator. It is easy to see that the
structure sheaves $\cO_{U_\alpha}$ and $\cO_{U_\beta}$ coincide in the
intersections $U_\alpha\cap U_\beta\,$; they can be glued together to
define an algebraic variety structure $(\bP^N_k,\cO_{\bP^N})$, such
that $\cO_{\bP^N,[z]}$ consists of all homogeneous
rational functions $p/q$ of degree $0$ (i.e., $\deg p=\deg q$),
such that $q(z)\ne 0$.

\claim{(1.30) Definition} An algebraic scheme or variety $(X,\cO_X)$
is said to be projective if it is isomorphic to a closed subscheme
of some projective space $(\bP^N_k,\cO_{\bP^N})$.
\endclaim

We now indicate a standard way of constructing projective schemes.
Let $S$ be a collection of homogeneous polynomials $P_j\in
k[z_0,\ldots,z_N]$, of degree $d_j\in\bN$. We define an associated
{\it projective algebraic set}
$$\wt V(S)=\big\{[z_0:\ldots:z_N]\in\bP^N_k\,;\,P(z)=0,~\forall
P\in S\big\}.$$
Let $J$ be the {\it homogeneous ideal} of $k[z_0,\ldots,z_N]$ generated by $S$
(recall that an ideal $J$ is said to be homogeneous if $J=\bigoplus J_m$
is the direct sum of its homogeneous components, or
equivalently, if $J$ is generated by homogeneous elements). We have an
associated graded algebra
$$B=k[z_0,\ldots,z_N]/J=\bigoplus B_m,\qquad B_m=k[z_0,\ldots,z_N]_m/J_m$$
such that $B$ is generated by $B_1$ and $B_m$ is a finite dimensional
vector space over $k$ for each~$k$. This is enough to construct the
desired scheme structure on $\wt V(J):=\bigcap\wt V(J_m)$, as we see
in the next subsection.

\subsubsection{1.D.6}{Projective Scheme Associated with a Graded Algebra.}
Let us start with a reduced graded $k$-algebra
$$B=\bigoplus_{m\in\bN}B_m$$
such that $B$ is generated by $B_0$ and $B_1$ as an algebra, and $B_0$,
$B_1$ are finite dimensional vector spaces over $k$ (it then follows
that $B$ is finitely generated and that all $B_m$ are finite
dimensional vector spaces). Given $s\in B_m$, $m>0$, we define
a $k$-algebra $A_s$ to be the ring of all fractions of homogeneous
degree~$0$ with a power of $s$ as their denominator, i.e.,
$$A_s=\big\{p/s^d\,;\,p\in B_{dm},\,d\in\bN\big\}.\leqno(1.31)$$
Since $A_s$ is generated by ${1\over s}B_1^m$ over $B_0$, $A_s$ is a
finitely generated algebra. We define $U_s=\Spm(A_s)$ to be the associated
affine algebraic variety. For $s\in B_m$ and $s'\in B_{m'}$, we clearly
have algebra homomorphisms
$$A_s\to A_{ss'},\qquad A_{s'}\to A_{ss'},$$
since $A_{ss'}$ is the algebra of all $0$-homogeneous fractions with
powers of $s$ and $s'$ in the denominator.
As $A_{ss'}$ is the same as the localized ring $A_s[s^{m'}/s^{\prime m}]$,
we see that $U_{ss'}$ can be identified with an affine open set in $U_s$,
and we thus get canonical injections
$$U_{ss'}\hookrightarrow U_s,\qquad U_{ss'}\hookrightarrow U_{s'}.$$

\claim{(1.32) Definition} If $B=\bigoplus_{m\in\bN}B_m$ is a reduced graded
algebra generated by its finite dimensional vector subspaces $B_0$ and $B_1$,
we associate an algebraic scheme $(X,\cO_X)=\Proj(B)$ as follows. To each
finitely generated algebra $A_s=\big\{p/s^d\,;\,p\in B_{dm},\,d\in\bN\big\}$
we associate an affine algebraic variety $U_s=\Spm(A_s)$. We let $X$ be the
union of all open charts $U_s$ with the identifications
$U_s\cap U_{s'}=U_{ss'}\,$; then the collection $(U_s)$ is a basis of the
topology of $X$, and $\cO_X$ is the unique sheaf of local $k$-algebras
such that $\Gamma(U_s,\cO_X)=A_s$ for each~$U_s$.
\endclaim

The following proposition shows that only finitely many open charts
are actually needed to describe~$X$ (as required in Def.~1.29~a)).

\claim{(1.33) Lemma} If $s_0,\ldots,s_N$ is a basis of $B_1$, then
$\Proj(B)=\bigcup\limits_{0\le j\le N}U_{s_j}$.
\endclaim

\proof{Proof} In fact, if $x\in X$ is contained in a chart $U_s$ for some
$s\in B_m$, then $U_s=\Spm(A_s)\ne\emptyset$, and therefore $A_s\ne\{0\}$.
As $A_s$ is generated by ${1\over s}B_1^m$, we can find a fraction
$f=s_{j_1}\ldots s_{j_m}/s$ representing an element $f\in\cO(U_s)$
such that $f(x)\ne 0$. Then $x\in U_s\ssm f^{-1}(0)$, and
$U_s\ssm f^{-1}(0)=\Spm(A_s[1/f])=U_s\cap U_{s_{j_1}}\cap\ldots\cap
U_{s_{j_m}}$. In particular $x\in U_{s_{j_1}}$.\qed
\endproof

\claim{(1.34) Example} \rm One can consider the {\it projective space}
$\bP^N_k$ to be the algebraic scheme
$$\bP^N_k=\Proj(k[T_0,\ldots,T_N]).$$
\endclaim

The Proj construction is fonctorial in the following sense: if we have
a graded homomorphism $\Phi:B\to B'$ (i.e.\ an algebra homomorphism
such that $\Phi(B_m)\subset B'_m$, then there are corresponding
morphisms $A_s\to A'_{\Phi(s)}$, $U'_{\Phi(s)}\to U_s$, and we thus find
a scheme morphism
$$F:\Proj(B')\to\Proj(B).$$
Also, since $p/s^d=ps^l/s^{d+l}$, the algebras $A_s$ depend only on
components $B_m$ of large degree, and we have $A_s=A_{s^l}$. It follows
easily that there is a canonical isomorphism
$$\Proj(B)\simeq\Proj\Big(\bigoplus_m B_{lm}\Big).$$ Similarly, we may
if we wish change a finite number of components $B_m$ without affecting
$\Proj(B)$. In particular, we may alway assume that $B_0=k\,1_B$.
By selecting finitely many generators $g_0,\ldots,g_N$ in $B_1$, we then
find a surjective graded homomorphism $k[T_0,\ldots,T_N]\to B$, thus
$B\simeq k[T_0,\ldots,T_N]/J$ for some graded ideal $J\subset B$. The algebra
homomorphism $k[T_0,\ldots,T_N]\to B$ therefore yields a scheme embedding
$\Proj(B)\to\bP^N$ onto $V(J)$.

We will not pursue further the study of algebraic varieties from this point
of view$\,$; in fact we are mostly interested in the case $k=\bC$, and
algebraic varieties over $\bC$ are a special case of the more general
concept of complex analytic space.

\section{2}{The Local Ring of Germs of Analytic Functions}
\subsection{2.A}{The Weierstrass Preparation Theorem}
Our first goal is to establish a basic factorization and division theorem
for analytic functions of several variables, which is essentially due to
Weierstrass. We follow here a simple proof given by C.L. Siegel, based on
a clever use of the Cauchy formula. Let $g$ be a holomorphic function
defined on a neighborhood of $0$ in $\bC^n$, $g\not\equiv 0$. There exists
a dense set of vectors $v\in\bC^n\ssm\{0\}$ such that the function
$\bC\ni t\longmapsto g(tv)$ is not identically zero. In fact the Taylor 
series of $g$ at the origin can be written
$$g(tv)=\sum_{k=0}^{+\infty}{1\over k!}\,t^k\,g^{(k)}(v)$$
where $g^{(k)}$ is a homogeneous polynomial of degree $k$ on $\bC^n$ and
$g^{(k_0)}\not\equiv 0$ for some index $k_0$. Thus it suffices to select
$v$ such that $g^{(k_0)}(v)\ne 0$. After a change of coordinates, we may assume
that $v=(0,\ldots,0,1)$. Let $s$ be the vanishing order of $z_n\longmapsto
g(0,\ldots,0,z_n)$ at $z_n=0$. There exists $r_n>0$ such that $g(0,\ldots,0,z_n)\ne 0$
when $0<|z_n|\le r_n$. By continuity of $g$ and compactness of the circle
$|z_n|=r_n$, there exists $r'>0$ and $\varepsilon>0$ such that
$$g(z',z_n)\ne 0~~~~\hbox{\rm for}~z'\in\bC^{n-1},
~~~|z'|\le r',~~~r_n-\varepsilon\le|z_n|\le r_n+\varepsilon.$$
For every integer $k\in\bN$, let us consider the integral
$$S_k(z')={1\over 2\pi\ii}\int_{|z_n|=r_n}{1\over g(z',z_n)}{\partial g\over
\partial z_n}(z',z_n)\,z_n^k\,dz_n.$$
Then $S_k$ is holomorphic in a neighborhood of $|z'|\le r'$. 
Rouch\'e's theorem shows that $S_0(z')$
is the number of roots $z_n$ of $g(z',z_n)=0$ in the disk $|z_n|<r_n$, thus
by continuity $S_0(z')$ must be a constant $s$. Let us denote by
$w_1(z'),\ldots,w_s(z')$ these roots, counted with multiplicity. By definition
of~$r_n$, we have $w_1(0)=\ldots=w_s(0)=0$, and by the choice of~$r'$,
$\varepsilon$ we have $|w_j(z')|<r_n-\varepsilon$ for $|z'|\le r'$.
The Cauchy residue formula yields
$$S_k(z')=\sum_{j=1}^s w_j(z')^k.$$
Newton's formula shows that the elementary symmetric function $c_k(z')$ of 
degree $k$ in $w_1(z'),\ldots,w_s(z')$ is a polynomial in $S_1(z'),\ldots,S_k(z')$.
Hence $c_k(z')$ is holomorphic in a neighborhood of $|z'|\le r'$. Let us set
$$P(z',z_n)=z_n^s-c_1(z')z_n^{s-1}+\cdots+(-1)^sc_s(z')=\prod_{j=1}^s
\big(z_n-w_j(z')\big).$$
For $|z'|\le r'$, the quotient $f=g/P$ (resp.\ $f=P/g$) is holomorphic in $z_n$
on the disk $|z_n|<r_n+\varepsilon$, because $g$ and $P$ have the same zeros
with the same multiplicities, and $f(z',z_n)$ is holomorphic in $z'$ for
$r_n-\varepsilon\le|z_n|\le r_n+\varepsilon$. Therefore
$$f(z',z_n)={1\over 2\pi\ii}\int_{|w_n|=r_n+\varepsilon}
{f(z',w_n)\,dw_n\over w_n-z_n}$$
is holomorphic in $z$ on a neighborhood of the closed polydisk 
$\ol\Delta(r',r_n)=\{|z'|\le r'\}\times\{|z_n|\le r_n\}$. 
Thus $g/P$ is invertible and we obtain:

\claim{(2.1) Weierstrass preparation theorem} Let $g$ be holomorphic
on a neighborhood of $0$ in $\bC^n$, such that $g(0,z_n)/z_n^s$ has
a not zero finite limit at $z_n=0$. With the above choice of $r'$ and $r_n$,
one can write $g(z)=u(z)P(z',z_n)$ where $u$ is an invertible holomorphic
function in a neighborhood of the polydisk $\ol\Delta(r',r_n)$, and $P$ is
a Weierstrass polynomial in~$z_n$, that is, a polynomial of the form
$$P(z',z_n)=z_n^s+a_1(z')z_n^{s-1}+\cdots+a_s(z'),~~~~a_k(0)=0,$$
with holomorphic coefficients $a_k(z')$ on a neighborhood of $|z'|\le r'$
in~$\bC^{n-1}$.
\endclaim

\claim{(2.2) Remark} \rm If $g$ vanishes at order $m$ at $0$ and $v\in\bC^n
\ssm\{0\}$ is selected such that $g^{(m)}(v)\ne 0$, then $s=m$ and $P$ must
also vanish at order $m$~at~$0$. In that case, the coefficients $a_k(z')$
are such that $a_k(z')=O(|z'|^k)$, $1\le k\le s$.
\endclaim

\claim{(2.3) Weierstrass division theorem} Every bounded 
holo\-morphic function $f$ on $\Delta=\Delta(r',r_n)$ can be represented in 
the form
$$f(z)=g(z)q(z)+R(z',z_n),\leqno(2.4)$$
where $q$ and $R$ are analytic in $\Delta$, $R(z',z_n)$ is a polynomial of
degree $\le s-1$ in $z_n$, and
$$\sup_\Delta|q|\le C\sup_\Delta|f|,~~~~\sup_\Delta|R|\le C\sup_\Delta|f|
\leqno(2.5)$$
for some constant $C\ge 0$ independent of $f$. The representation $(2.4)$
is unique.
\endclaim

\proof{Proof {\rm(Siegel)}} It is sufficient to prove the result when 
$g(z)=P(z',z_n)$ is a Weierstrass polynomial. 

Let us first prove the uniqueness. If $f=Pq_1+R_1=Pq_2+R_2$, then
$$P(q_2-q_1)+(R_2-R_1)=0.$$
It follows that the $s$ roots $z_n$ of $P(z',\bu)=0$ are zeros of $R_2-R_1$. 
Since deg$_{z_n}(R_2-R_1)\le s-1$, we must have $R_2-R_1\equiv 0$, thus 
$q_2-q_1\equiv 0$.

In order to prove the existence of $(q,R)$, we set
$$q(z',z_n)=\lim_{\varepsilon\to 0{\scriptscriptstyle+}}~{1\over 2\pi\ii}
\int_{|w_n|=r_n-\varepsilon}{f(z',w_n)\over P(z',w_n)(w_n-z_n)}\,dw_n,
~~~~z\in\Delta~;$$
observe that the integral does not depend on $\varepsilon$ when
$\varepsilon<r_n-|z_n|$ is small enough. Then $q$ is holomorphic on
$\Delta$. The function $R=f-Pq$ is also holomorphic on $\Delta$ and
$$R(z)=\lim_{\varepsilon\to 0{\scriptscriptstyle+}}~{1\over 2\pi\ii}
\int_{|w_n|=r_n-\varepsilon}{f(z',w_n)\over P(z',w_n)}\,
\Big[{P(z',w_n)-P(z',z_n)\over(w_n-z_n)}\Big]\,dw_n.$$
The expression in brackets has the form
$$\big[(w_n^s-z_n^s)+\sum_{j=1}^s a_j(z')(w_n^{s-j}-z_n^{s-j})\big]/(w_n-z_n)$$
hence is a polynomial in $z_n$ of degree $\le s-1$ with coefficients that are
holomorphic functions of $z'$. Thus we have the asserted decomposition
$f=Pq+R$ and
$$\sup_\Delta|R|\le C_1\sup_\Delta|f|$$
where $C_1$ depends on bounds for the $a_j(z')$ and on $\mu=\min|P(z',z_n)|$
on the compact set $\{|z'|\le r'\}\times\{|z_n|=r_n\}$. By the maximum 
principle applied to $q=(f-R)/P$ on each disk $\{z'\}\times
\{|z_n|<r_n-\varepsilon\}$, we easily get
$$\sup_\Delta|q|\le\mu^{-1}(1+C_1)\sup_\Delta|f|.\eqno\square$$
\endproof

\subsection{2.B}{Algebraic Properties of the Ring $\cO_n$}
We give here important applications of the Weierstrass preparation theorem
to the study of the ring of germs of holomorphic functions in $\bC^n$.

\claim{(2.6) Notation} We let $\cO_n$ be the ring of germs of
holomorphic functions on $\bC^n$ at $0$. Alternatively, $\cO_n$ can
be identified with the ring $\bC\{z_1,\ldots,z_n\}$ of convergent power
series in $z_1,\ldots,z_n$. \endclaim

\claim{(2.7) Theorem} The ring $\cO_n$ is Noetherian, i.e.\ every ideal
$\cI$ of $\cO_n$ is finitely gene\-rated.
\endclaim

\proof{Proof} By induction on $n$. For $n=1$, $\cO_n$ is principal:
every ideal $\cI\ne\{0\}$ is generated by $z^s$, where $s$ is the minimum
of the vanishing orders at $0$ of the non zero elements of $\cI$.
Let $n\ge 2$ and $\cI\subset\cO_n$, $\cI\ne\{0\}$. After a change of 
variables, we may assume that $\cI$ contains a Weierstrass polynomial
$P(z',z_n)$. For every $f\in\cI$, the Weierstrass division theorem yields
$$f(z)=P(z',z_n)q(z)+R(z',z_n),~~~~R(z',z_n)=\sum_{k=0}^{s-1}c_k(z')\,z_n^k,$$
and we have $R\in\cI$. Let us consider the set $\cM$ of coefficients
$(c_0,\ldots,c_{s-1})$ in $\cO_{n-1}^{\oplus s}$ corresponding to the polynomials
$R(z',z_n)$ which belong to $\cI$. Then $\cM$ is a $\cO_{n-1}$-submodule of
$\cO_{n-1}^{\oplus s}$. By the induction hypothesis $\cO_{n-1}$ is Noetherian;
furthermore, every submodule of a finitely generated module over a Noetherian 
ring is finitely generated (\cite{Lang~1965}, Chapter~VI). Therefore $\cM$ is
finitely generated, and $\cI$ is generated by $P$ and by polynomials
$R_1,\ldots,R_N$ associated with a finite set of generators of~$\cM$.\qed
\endproof

Before going further, we need two lemmas which relate the algebraic properties
of $\cO_n$ to those of the polynomial ring $\cO_{n-1}[z_n]$.

\claim{(2.8) Lemma} Let $P,F\in\cO_{n-1}[z_n]$ where $P$ is a Weierstrass 
polynomial. If $P$ divides $F$ in $\cO_n$, then $P$ divides $F$ in
$\cO_{n-1}[z_n]$.
\endclaim

\proof{Proof} Assume that $F(z',z_n)=P(z',z_n)h(z)$, $h\in\cO_n$. The
standard division algorithm of $F$ by $P$ in $\cO_{n-1}[z_n]$ yields
$$F=PQ+R,~~~~Q,R\in\cO_{n-1}[z_n],~~~\deg\,R<\deg\,P.$$
The uniqueness part of Th.~2.3 implies $h(z)=Q(z',z_n)$ and 
$R\equiv 0$.\qed
\endproof

\claim{(2.9) Lemma} Let $P(z',z_n)$ be a Weierstrass polynomial.
\smallskip
\item{\rm a)} If $P=P_1\ldots P_N$ with $P_j\in\cO_{n-1}[z_n]$, then, up to
invertible elements of $\cO_{n-1}$, all $P_j$ are Weierstrass
polynomials.\smallskip
\item{\rm b)} $P(z',z_n)$ is irreducible in $\cO_n$ if and only if it is 
irreducible in $\cO_{n-1}[z_n]$.
\vskip0pt
\endclaim

\proof{Proof} a) Assume that $P=P_1\ldots P_N$ with polynomials
$P_j\in\cO_{n-1} [z_n]$ of respective degrees $s_j$, $\sum_{1\le j\le
N}s_j=s$. The product  of the leading coefficients of $P_1,\ldots,P_N$ in
$\cO_{n-1}$ is equal to~$1$; after normalizing these polynomials, we
may assume that $P_1,\ldots,P_N$ are unitary and $s_j>0$ for all~$j$. Then
$$P(0,z_n)=z_n^s=P_1(0,z_n)\ldots P_N(0,z_n),$$
hence $P_j(0,z_n)=z_n^{s_j}$ and therefore $P_j$ is a Weierstrass polynomial.
\smallskip

\noindent b) Set $s=\deg\,P$ and $P(0,z_n)=z_n^s$. Assume that $P$ is
reducible in $\cO_n$, with $P(z',z_n)=g_1(z)g_2(z)$ for non invertible
elements $g_1,g_2\in\cO_n$. Then $g_1(0,z_n)$ and $g_2(0,z_n)$ have
vanishing orders $s_1,s_2>0$ with $s_1+s_2=s$, and
$$g_j=u_jP_j,~~~~\deg\,P_j=s_j,~~~j=1,2,$$ where $P_j$ is a Weierstrass
polynomial and $u_j\in\cO_n$ is invertible. Therefore $P_1P_2=uP$ for
an invertible germ $u\in\cO_n$. Lemma~2.8 shows that $P$ divides
$P_1P_2$ in $\cO_{n-1}[z_n]\,$; since $P_1$, $P_2$  are unitary and
$s=s_1+s_2$, we get $P=P_1P_2$, hence $P$ is reducible in
$\cO_{n-1}[z_n]$. The converse implication is obvious from~a).
\qed
\endproof

\claim{(2.10) Theorem} $\cO_n$ is a factorial ring, i.e.\ $\cO_n$ is
entire and:
\smallskip
\item{\rm a)} every non zero germ $f\in\cO_n$ admits a factorization 
$f=f_1\ldots f_N$ in irreducible elements$\,;$
\smallskip
\item{\rm b)} the factorization is unique up to invertible elements.
\endclaim

\proof{Proof} The existence part a) follows from Lemma~2.9 if we take
$f$ to be a Weierstrass polynomial and $f=f_1\ldots f_N$ be a decomposition
of maximal length $N$ into polynomials of positive degree. In order to prove
the uniqueness, it is sufficient to verify the following statement:
\smallskip
\item{\rlap{\hbox{${\rm b}')$}}} {\it If $g$ is an irreducible element that
divides a product $f_1f_2$, then $g$ divides either $f_1$ or $f_2$.}
\smallskip
\noindent By Th.~2.1, we may assume that $f_1$, $f_2$, $g$ are 
Weierstrass polynomials in $z_n$. Then $g$ is irreducible and divides
$f_1f_2$ in $\cO_{n-1}[z_n]$ thanks to Lemmas 2.8 and 2.9~b).
By induction on $n$, we may assume that $\cO_{n-1}$ is factorial. 
The standard Gauss lemma (\cite{Lang 1965}, Chapter V) says that the
polynomial ring $A[T]$ is factorial if the ring $A$ is
factorial. Hence $\cO_{n-1}[z_n]$ is factorial by induction and thus
$g$ must divide $f_1$ or $f_2$ in $\cO_{n-1}[z_n]$.\qed
\endproof

\claim{(2.11) Theorem} If $f,g\in\cO_n$ are relatively prime, then
the germs $f_z,~g_z$ at every point $z\in\bC^n$ near $0$ are again
relatively prime.
\endclaim

\proof{Proof} One may assume that $f=P,~g=Q$ are Weierstrass polynomials. 
Let us recall that unitary polynomials $P,Q\in\cA[X]$ ($\cA={\rm a}$
factorial ring) are relatively prime if and only if their resultant 
$R\in\cA$ is non zero. Then the resultant $R(z')\in\cO_{n-1}$ of
$P(z',z_n)$ and $Q(z',z_n)$ has a non zero germ at $0$. Therefore the germ 
$R_{z'}$ at points $z'\in\bC^{n-1}$ near $0$ is also non zero.\qed
\endproof

\section{3}{Coherent Sheaves}
\subsection{3.1}{Locally Free Sheaves and Vector Bundles}
Section~9 will
greatly develope this philosophy. Before introducing the more general
notion of a coherent sheaf, we discuss the notion of locally free
sheaves over a sheaf a ring. All rings occurring in the sequel are
supposed to be commutative with unit (the non commutative case is also
of considerable interest, e.g.\ in view of the theory of $\cD$-modules,
but this subject is beyond the scope of the present book).

\claim{(3.1) Definition} Let $\cA$ be a sheaf of rings on a
topological space $X$ and let $\cS$ a sheaf of modules over $\cA$ (or
briefly a $\cA$-module). Then $\cS$ is said to be locally free
of rank $r$ over $\cA$, if $\cS$ is locally isomorphic to $\cA^{\oplus r}$
on a neighborhood of every point, i.e.\ for every $x_0\in X$ one can find
a neighborhood $\Omega$ and sections $F_1,\ldots,F_r\in\cS(\Omega)$ such that
the sheaf homomorphism
$$F:\cA_{\restriction\Omega}^{\oplus r}\longrightarrow\cS_{\restriction\Omega},~~~~
\cA_x^{\oplus r}\ni(w_1,\ldots,w_r)\longmapsto
\sum_{1\le j\le r}w_jF_{j,x}\in\cS_x$$
is an isomorphism.
\endclaim

By definition, if $\cS$ is locally free, there is a covering
$(U_\alpha)_{\alpha\in I}$ by open sets on which $\cS$ admits free
generators $F_\alpha^1,\ldots,F_\alpha^r\in\cS(U_\alpha)$. Because the
generators can be uniquely expressed in terms of any other system of
independent generators, there is for each pair $(\alpha,\beta)$ a
$r\times r$ matrix
$$G_{\alpha\beta}=(G_{\alpha\beta}^{jk})_{1\le j,k\le r},\qquad
G_{\alpha\beta}^{jk}\in\cA(U_\alpha\cap U_\beta),$$
such that 
$$F_\beta^k=\sum_{1\le j\le r}F_\alpha^jG^{jk}_{\alpha\beta}\qquad
\hbox{\rm on}\quad U_\alpha\cap U_\beta.$$
In other words, we have a commutative diagram
$$\eqalign{
&\cA^{\oplus r}_{\restriction U_\alpha\cap U_\beta}
\buildo{\displaystyle F_\alpha}\over\longrightarrow
\cS_{\restriction U_\alpha\cap U_\beta}\cr
&G_{\alpha\beta}\Big\uparrow\quad\phantom{\longrightarrow}\qquad
\Big|~\Big|\cr
&\cA^{\oplus r}_{\restriction U_\alpha\cap U_\beta}
\buildu{\displaystyle F_\beta}\under\longrightarrow
\cS_{\restriction U_\alpha\cap U_\beta}\cr}$$
It follows easily from the equality $G_{\alpha\beta}=F_\alpha^{-1}\circ
F_\beta$ that the {\it transition matrices} $G_{\alpha\beta}$ are invertible
matrices satisfying the transition relation
$$G_{\alpha\gamma}=G_{\alpha\beta}G_{\beta\gamma}\qquad\hbox{\rm on}\quad
U_\alpha\cap U_\beta\cap U_\gamma\leqno(3.2)$$
for all indices $\alpha,\beta,\gamma\in I$. In particular $G_{\alpha\alpha}=
\Id$ on $U_\alpha$ and $G_{\alpha\beta}^{-1}=G_{\beta\alpha}$ on
$U_\alpha\cap U_\beta$.

Conversely, if we are given a system of invertible $r\times r$ matrices
$G_{\alpha\beta}$ with coefficients in $\cA(U_\alpha\cap U_\beta)$
satisfying the transition relation~(3.2), we can define a locally free
sheaf $\cS$ of rank $r$ over $\cA$ by taking $\cS\simeq\cA^{\oplus r}$
over each $U_\alpha$, the identification over $U_\alpha\cap U_\beta$ being
given by the isomorphism $G_{\alpha\beta}$. A~section $H$ of $\cS$ over an
open set $\Omega\subset X$ can just be seen as a collection of sections
$H_\alpha=(H_\alpha^1,\ldots,H_\alpha^r)$ of $\cA^{\oplus r}(\Omega\cap
U_\alpha)$ satisfying the transition relations $H_\alpha=G_{\alpha\beta}
H_\beta$ over $\Omega\cap U_\alpha\cap U_\beta$.

The notion of locally free sheaf is closely related to another essential
notion of differential geometry, namely the notion of vector bundle
(resp.\ topological, differentiable, holomorphic $\ldots$, vector bundle).
To describe the relation between these notions, we assume that the sheaf
of rings $\cA$ is a subsheaf of the sheaf $\cC_\bK$ of continous functions
on $X$ with values in the field $\bK=\bR$ or $\bK=\bC$, containing
the sheaf of locally constant functions $X\to\bK$. Then, for each $x\in X$,
there is an evaluation map
$$\cA_x\to\bK,\qquad w\mapsto w(x)$$
whose kernel is a maximal ideal $\gm_x$ of $\cA_x$, and $\cA_x/\gm_x=\bK$.
Let $\cS$ be a locally free sheaf of rank $r$ over~$\cA$. To each
$x\in X$, we can associate a $\bK$-vector space $E_x=\cS_x/\gm_x\cS_x$:
since $\cS_x\simeq\cA_x^{\oplus r}$, we have
$E_x\simeq(\cA_x/\gm_x)^{\oplus r}=\bK^r$. The set
$E=\coprod_{x\in X} E_x$ is equipped with a natural projection
$$\pi:E\to X,\qquad \xi\in E_x\mapsto \pi(\xi):=x,$$
and the fibers $E_x=\pi^{-1}(x)$ have a structure of $r$-dimensional
$\bK$-vector space: such a structure $E$ is called a {\it $\bK$-vector
bundle of rank $r$} over~$X$. Every section $s\in \cS(U)$ gives rise to
a {\it section} of $E$ over $U$ by setting $s(x)=s_x$~mod~$\gm_x$. We
obtain a function (still denoted by the same symbol) $s:U\to E$ such that
$s(x)\in E_x$ for every $x\in U$, i.e.\ $\pi\circ s=\Id_U$. It is clear
that $\cS(U)$ can be considered as a $\cA(U)$-submodule of the
$\bK$-vector space of functions $U\to E$ mapping a point $x\in U$ to
an element in the fiber~$E_x$. Thus we get
a subsheaf of the sheaf of $E$-valued sections, which is in a natural way
a $\cA$-module isomorphic to~$\cS$. This subsheaf will be denoted by
$\cA(E)$ and will be called the {\it sheaf of $\cA$-sections} of~$E$.
If we are given a $\bK$-vector bundle $E$ over $X$ and a subsheaf
$\cS=\cA(E)$ of the sheaf of all sections of $E$ which is in a natural way
a locally free $\cA$-module of rank~$r$, we say that $E$ (or more
precisely the pair $(E,\cA(E))$) is a $\cA$-vector bundle of rank~$r$
over~$X$. 

\claim{(3.3) Example} \rm In case $\cA=\cC_{X,\bK}$ is the sheaf of all
$\bK$-valued continuous functions on~$X$, we say that $E$ is a
{\it topological} vector bundle over~$X$. When $X$ is a manifold and
$\cA=\cC^p_{X,\bK}$, we say that $E$ is a {\it $C^p$-differentiable}
vector bundle; finally, when $X$ is complex analytic and $\cA=\cO_X$, we say
that $E$ is a {\it holomorphic} vector bundle.
\endclaim

Let us introduce still a little more notation. Since $\cA(E)$ is a locally
free sheaf of rank $r$ over any open set $U_\alpha$ in a suitable covering
of $X$, a choice of generators $(F_\alpha^1,\ldots,F_\alpha^r)$ for
$\cA(E)_{\restriction U_\alpha}$ yields corresponding generators
$(e_\alpha^1(x),\ldots,e_\alpha^r(x))$ of the fibers~$E_x$ over~$\bK$. Such a
system of gene\-rators is called a {\it $\cA$-admissible frame} of $E$
over~$U_\alpha$. There is a corresponding isomorphism
$$\theta_\alpha:E_{\restriction U_\alpha}:=\pi^{-1}(U_\alpha)\longrightarrow
U_\alpha\times\bK^r\leqno(3.4)$$
which to each $\xi\in E_x$ associates the pair
$(x,(\xi_\alpha^1,\ldots,\xi_\alpha^r))\in U_\alpha\times\bK^r$ composed of $x$
and of the components $(\xi_\alpha^j)_{1\le j\le r}$ of $\xi$ in the basis
$(e_\alpha^1(x),\ldots,e_\alpha^r(x))$ of~$E_x$. The bundle $E$ is said to
be {\it trivial} if it is of the form $X\times\bK^r$, which is the same
as saying that $\cA(E)=\cA^{\oplus r}$. For this reason, the isomorphisms
$\theta_\alpha$ are called {\it trivializations} of $E$ over~$U_\alpha$.
The corresponding {\it transition automorphisms} are
$$\eqalign{
&\theta_{\alpha\beta}:=\theta_\alpha\circ\theta_\beta^{-1}:
(U_\alpha\cap U_\beta)\times \bK^r\longrightarrow
(U_\alpha\cap U_\beta)\times \bK^r,\cr
&\theta_{\alpha\beta}(x,\xi)=(x,g_{\alpha\beta}(x)\cdot \xi),
\qquad (x,\xi)\in (U_\alpha\cap U_\beta)\times \bK^r,\cr}$$
where $(g_{\alpha\beta})\in\GL_r(\cA)(U_\alpha\cap U_\beta)$ are the
transition matrices already described (except that they are just seen as
matrices with coefficients in $\bK$ rather than with coefficients in
a sheaf). Conversely, if we are given a collection of matrices
$g_{\alpha\beta}=(g_{\alpha\beta}^{jk})\in\GL_r(\cA)(U_\alpha\cap U_\beta)$ satisfying the
transition relation
$$g_{\alpha\gamma}=g_{\alpha\beta}g_{\beta\gamma}\qquad\hbox{\rm on}\quad
U_\alpha\cap U_\beta\cap U_\gamma,$$
we can define a $\cA$-vector bundle
$$E=\big(\coprod_{\alpha\in I}U_\alpha\times\bK^r\big)/\sim$$
by gluing the charts $U_\alpha\times\bK^r$ via the identification
$(x_\alpha,\xi_\alpha)\sim(x_\beta,\xi_\beta)$ if and only 
if $x_\alpha=x_\beta=x\in U_\alpha\cap U_\beta$ and
$\xi_\alpha=g_{\alpha\beta}(x)\cdot\xi_\beta$.

\claim{(3.5) Example} \rm When $X$ is a real differentiable manifold,
an interesting example of real vector bundle 
is the {\it tangent bundle} $T_X$~; if $\tau_\alpha:U_\alpha\to\bR^n$ 
is a collection of coordinate charts on $X$, then 
\hbox{$\theta_\alpha=\pi\times d\tau_\alpha:T_{X\restriction U_\alpha}\to 
U_\alpha\times\bR^m$} define trivializations of $T_X$ 
and the transition matrices are given by $g_{\alpha\beta}(x)=
d\tau_{\alpha\beta}(x^\beta)$ where $\tau_{\alpha\beta}=
\tau_\alpha\circ\smash{\tau_\beta^{-1}}$ and $x^\beta=\tau_\beta(x)$.
The dual $T^\star_X$ of $T_X$ is called the {\it cotangent bundle} of~$X$.
If $X$ is complex analytic, then $T_X$ has the structure of a holomorphic
vector bundle.
\endclaim

We now briefly discuss the concept of sheaf and bundle morphisms. If
$\cS$ and $\cS'$ are sheaves of $\cA$-modules over a topological space~$X$,
then by a morphism $\varphi:\cS\to\cS'$ we just mean a $\cA$-linear
sheaf morphism. If $\cS=\cA(E)$ and $\cS'=\cA(E')$ are locally free
sheaves, this is the same as a $\cA$-linear bundle morphism, that is,
a fiber preserving $\bK$-linear morphism $\varphi(x):E_x\to E'_x$
such that the matrix representing $\varphi$ in any local $\cA$-admissible
frames of $E$ and $E'$ has coefficients in~$\cA$.

\claim{(3.6) Proposition} Suppose that $\cA$ is a sheaf of local
rings, i.e.\ that a section of $\cA$ is invertible in $\cA$ if and only
if it never takes the zero value in~$\bK$. Let $\varphi:\cS\to \cS'$
be a $\cA$-morphism of locally free $\cA$-modules of rank $r$,~$r'$. If
the rank of the $r'\times r$ matrix $\varphi(x)\in M_{r'r}(\bK)$
is constant for all $x\in X$, then $\Ker\varphi$ and $\Im\varphi$
are locally free subsheaves of $\cS$, $\cS'$ respectively, and
$\Coker\varphi=\cS'/\Im\varphi$ is locally free.
\endclaim

\proof{Proof} This is just a consequence of elementary linear algebra, once we
know that non zero determinants with coefficients in $\cA$ can be inverted.
\qed
\endproof

Note that all three sheaves $\cC_{X,\bK}$, $\cC^p_{X,\bK}$, $\cO_X$
are sheaves of local rings, so Prop.~3.6 applies to these cases. However,
even if we work in the holomorphic category ($\cA=\cO_X$), a difficulty
immediately appears that the kernel or cokernel of an arbitrary morphism
of locally free sheaves is in general not locally free.

\claim{(3.7) Examples} {\rm\smallskip
\item{a)} Take $X=\bC$, let $\cS=\cS'=\cO$ be the trivial sheaf, and let
$\varphi:\cO\to\cO$ be the morphism $u(z)\mapsto z\,u(z)$. It is immediately
seen that $\varphi$ is injective as a sheaf morphism ($\cO$ being an entire
ring), and that $\Coker\varphi$ is the {\it skyscraper sheaf} $\bC_0$
of stalk $\bC$ at $z=0$, having zero stalks at all other points $z\ne 0$.
Thus $\Coker\varphi$ is not a locally free sheaf, although $\varphi$ is
everywhere injective (note however that the corresponding morphism \hbox{
$\varphi:E\to E'$}, $(z,\xi)\mapsto (z,z\xi)$ of trivial rank~$1$ vector
bundles $E=E'=\bC\times\bC$ is {\it not injective} on the zero fiber
$E_0$).
\smallskip
\item{b)} Take $X=\bC^3$, $\cS=\cO^{\oplus 3}$, $\cS'=\cO$ and
$$\varphi:\cO^{\oplus 3}\to\cO,\qquad (u_1,u_2,u_3)\mapsto
\sum_{1\le j\le 3}z_ju_j(z_1,z_2,z_3).$$
Since $\varphi$ yields a surjective bundle morphism on $\bC^3\ssm\{0\}$,
one easily sees that $\Ker\varphi$ is locally free of rank~$2$ over
$\bC^3\ssm\{0\}$. However, by looking at the Taylor expansion of the
$u_j$'s at $0$, it is not difficult to check that $\Ker\varphi$ is the
$\cO$-submodule of $\cO^{\oplus 3}$ generated by the three sections
$(-z_2,z_1,0)$, $(-z_3,0,z_1)$ and $(0,z_3,-z_2)$, and that any two of these
three sections cannot generate the $0$-stalk $(\Ker\varphi)_0$. Hence
$\Ker\varphi$ is not locally free.\smallskip}
\endclaim

Since the category of locally free $\cO$-modules is not stable by
taking kernels or cokernels, one is led to introduce a more general
category which will be stable under these operations. This leads to
the notion of {\it coherent sheaves}.

\subsection{3.2}{Notion of Coherence}
The notion of coherence again deals with sheaves of modules over a sheaf
of rings. It is a semi-local property which says roughly that the sheaf
of modules locally has a finite presentation in terms of generators and
relations. We describe here some general properties of this notion,
before concentrating ourselves on the case of coherent $\cO_X$-modules.

\claim{(3.8) Definition} Let $\cA$ be a sheaf of rings on a
topological space $X$ and $\cS$ a sheaf of modules over $\cA$ (or
briefly a $\cA$-module). Then $\cS$ is said to be locally finitely
generated if for every point $x_0\in X$ one can find a neighborhood
$\Omega$ and sections $F_1,\ldots,F_q\in\cS(\Omega)$ such that for every
$x\in\Omega$ the stalk $\cS_x$ is generated by the germs $F_{1,x},\ldots,
F_{q,x}$ as an $\cA_x$-module. \endclaim

\claim{(3.9) Lemma} Let $\cS$ be a locally finitely generated sheaf of
$\cA$-modules on $X$ and $G_1,\ldots,G_N$ sections in $\cS(U)$ such that
$G_{1,x_0},\ldots,G_{N,x_0}$ generate $\cS_{x_0}$ at $x_0\in U$.  Then
$G_{1,x},\ldots,G_{N,x}$ generate $\cS_x$ for $x$ near $x_0$.
\endclaim

\proof{Proof} Take $F_1,\ldots,F_q$ as in Def.~3.8. As $G_1,\ldots,G_N$
generate $\cS_{x_0}$, one can find
a neighborhood $\Omega'\subset\Omega$ of $x_0$ and $H_{jk}\in\cA(\Omega')$
such that $F_j=\sum H_{jk}G_k$ on $\Omega'$. Thus $G_{1,x},\ldots,G_{N,x}$
generate $\cS_x$ for all $x\in\Omega'$.\qed
\endproof

\subsubsection{3.2.1}{Definition of Coherent Sheaves.}
If $U$ is an open subset of $X$, we denote by $\cS_{\restriction U}$ the
restriction of $\cS$ to $U$, i.e.\ the union of all stalks $\cS_x$
for $x\in U$. If $F_1,\ldots,F_q\in\cS(U)$, the kernel of the sheaf homomorphism
$F:\cA_{\restriction U}^{\oplus q}\longrightarrow\cS_{\restriction U}$ defined by
$$\cA_x^{\oplus q}\ni(g^1,\ldots,g^q)\longmapsto\sum_{1\le j\le q}g^jF_{j,x}
\in\cS_x,~~~~x\in U\leqno(3.10)$$
is a subsheaf $\cR(F_1,\ldots,F_q)$ of $\cA_{\restriction U}^{\oplus q}$, called
the {\it sheaf of relations} between $F_1,\ldots,F_q$.

\claim{(3.11) Definition} A sheaf $\cS$ of $\cA$-modules on $X$ is said to be
coherent if:
\smallskip
\item{\rm a)} $\cS$ is locally finitely generated~$;$
\smallskip
\item{\rm b)} for any open subset $U$ of $X$ and any 
$F_1,\ldots,F_q\in\cS(U)$, the sheaf of relations
$\cR(F_1,\ldots,F_q)$ is locally finitely generated.\smallskip
\endclaim

Assumption a) means that every point $x\in X$ has a neighborhood
$\Omega$ such that there is a surjective sheaf morphism
$F:\cA_{\restriction\Omega}^{\oplus q}\longrightarrow \cS_{\restriction \Omega}$, and 
assumption b) implies that the kernel of $F$ is locally finitely generated.
Thus, after shrinking $\Omega$, we see that $\cS$ admits over $\Omega$
a finite presentation under the form of an exact sequence
$$\cA_{\restriction\Omega}^{\oplus p}\buildo G\over\longrightarrow\cA_{\restriction\Omega}^{\oplus q}
\buildo F\over\longrightarrow\cS_{\restriction\Omega}\longrightarrow 0,\leqno(3.12)$$
where $G$ is given by a $q\times p$ matrix $(G_{jk})$ of sections of
$\cA(\Omega)$ whose columns $(G_{j1})$, $\ldots\,$, $(G_{jp})$ are generators of
$\cR(F_1,\ldots,F_q)$.

It is clear that every locally finitely generated subsheaf of a
coherent sheaf is coherent. From this we easily infer:

\claim{(3.13) Theorem} Let $\varphi:\cF\longrightarrow\cG$ be a $\cA$-morphism of 
coherent sheaves. Then $\Im\varphi$ and $\ker\varphi$ are coherent.
\endclaim

\proof{Proof} Clearly $\Im\varphi$ is a locally finitely generated subsheaf
of $\cG$, so it is coherent. Let $x_0\in X$, let $F_1,\ldots,F_q\in\cF(\Omega)$
be generators of $\cF$ on a neighborhood $\Omega$ of $x_0$, and $G_1,\ldots,G_r\in
\cA(\Omega')^{\oplus q}$ be generators of $\cR\big(\varphi(F_1),\ldots,\varphi(F_q)\big)$
on a neighborhood $\Omega'\subset\Omega$ of $x_0$. Then $\ker\varphi$ is
generated over $\Omega'$ by the sections
$$H_j=\sum_{k=1}^q G_j^kF_k\in\cF(\Omega'),~~~~1\le j\le r.\eqno\square$$
\endproof

\claim{(3.14) Theorem} Let $0\longrightarrow\cF\longrightarrow\cS\longrightarrow\cG\longrightarrow 0$ be an exact
sequence of $\cA$-modules. If two of the sheaves $\cF,\cS,\cG$ are
coherent, then all three are coherent.
\endclaim

\proof{Proof} If $\cS$ and $\cG$ are coherent, then $\cF=\ker(\cS\to\cG)$ is
coherent by Th.~3.13. If $\cS$ and $\cF$ are coherent, then
$\cG$ is locally finitely generated; to prove the coherence, let 
$G_1,\ldots,G_q\in\cG(U)$ and $x_0\in U$. Then there is a neighborhood 
$\Omega$ of $x_0$ and sections $\tilde G_1,\ldots,\tilde G_q\in\cS(\Omega)$ 
which are mapped to $G_1,\ldots,G_q$ on $\Omega$. After shrinking $\Omega$, 
we may assume also that $\cF_{\restriction\Omega}$ is generated by 
sections $F_1,\ldots,F_p\in\cF(\Omega)$. Then $\cR(G_1,\ldots,G_q)$ is the 
projection on the last $q$-components of 
$\cR(F_1,\ldots,F_p,\tilde G_1,\ldots,\tilde G_q)\subset\cA^{p+q}$, which is 
finitely generated near $x_0$ by the coherence of $\cS$. Hence 
$\cR(G_1,\ldots,G_q)$ is finitely generated near $x_0$ and $\cG$ is coherent.

Finally, assume that $\cF$ and $\cG$ are coherent.  Let $x_0\in X$ be
any point, let $F_1,\ldots,F_p\in\cF(\Omega)$ and $G_1,\ldots,G_q\in\cG(\Omega)$
be generators of $\cF$, $\cG$ on a neighborhood $\Omega$ of $x_0$.  There
is a neighborhood $\Omega'$ of $x_0$ such that $G_1,\ldots,G_q$ admit liftings
$\tilde G_1,\ldots,\tilde G_q\in\cS(\Omega')$.  Then $(F_1,\ldots,F_q,\tilde
G_1,\ldots,\tilde G_q)$ generate $\cS_{\restriction\Omega'}$, so $\cS$ is
locally finitely generated.  Now, let $S_1,\ldots,S_q$ be arbitrary sections
in $\cS(U)$ and $\ol S_1,\ldots,\ol S_q$ their images in $\cG(U)$.  For any
$x_0\in U$, the sheaf of relations $\cR(\ol S_1,\ldots,\ol S_q)$ is generated
by sections $P_1,\ldots,P_s\in\cA(\Omega)^{\oplus q}$ on a small neighborhood $\Omega$
of $x_0$. Set $P_j=(P_j^k)_{1\le k\le q}$.  Then $H_j=P_j^1S_1+\ldots+P_j^qS_q$,
$1\le j\le s$, are mapped to $0$ in $\cG$ so they can be seen as
sections of $\cF$.  The coherence of $\cF$ shows that $\cR(H_1,\ldots,H_s)$
has generators $Q_1,\ldots,Q_t\in\cA(\Omega')^s$ on a small neighborhood
$\Omega'\subset\Omega$ of $x_0$.  Then $\cR(S_1,\ldots,S_q)$ is generated
over $\Omega'$ by $R_j=\sum Q_j^kP_k\in\cA(\Omega')$, $1\le j\le t$,
and $\cS$ is coherent.\qed
\endproof

\claim{(3.15) Corollary} If $\cF$ and $\cG$ are coherent subsheaves
of a coherent analytic sheaf $\cS$, the intersection $\cF\cap\cG$ is a
coherent sheaf.
\endclaim

\proof{Proof} Indeed, the intersection sheaf $\cF\cap\cG$ is the kernel of the 
composite morphism $\cF\lhra\cS\longrightarrow\cS/\cG,$ and $\cS/\cG$ is
coherent.\qed
\endproof

\subsubsection{3.2.2}{Coherent Sheaf of Rings.} A sheaf of rings $\cA$ is said
to be coherent if it is coherent as a module over itself. By Def.~3.11,
this means that for any open set $U\subset X$ and any sections
$F_j\in\cA(U)$, the sheaf of relations $\cR(F_1,\ldots,F_q)$ is finitely
generated. The above results then imply that all free modules $\cA^{\oplus p}$
are coherent. As a consequence:

\claim{(3.16) Theorem} If $\cA$ is a coherent sheaf of rings, any
locally finitely gene\-rated subsheaf of $\cA^{\oplus p}$ is coherent. In
particular, if $\cS$ is a coherent \hbox{\it$\cA$-module} 
and $F_1,\ldots,F_q\in\cS(U)$, the sheaf of relations 
$\cR(F_1,\ldots,F_q)\subset\cA^{\oplus q}$ is also coherent.
\endclaim

Let $\cS$ be a coherent sheaf of modules over a coherent sheaf of ring
$\cA$.  By an iteration of construction (3.12), we see that for
every integer $m\ge 0$ and every point $x\in X$ there is a neighborhood 
$\Omega$ of $x$ on which there is an exact sequence of sheaves
$$\cA_{\restriction\Omega}^{\oplus p_m}\buildo F_m\over\longrightarrow
\cA_{\restriction\Omega}^{\oplus p_{m-1}}\longrightarrow\cdots\longrightarrow
\cA_{\restriction\Omega}^{\oplus p_1}\buildo F_1\over\longrightarrow
\cA_{\restriction\Omega}^{\oplus p_0}\buildo F_0\over\longrightarrow
\cS_{\restriction\Omega}\longrightarrow 0,\leqno(3.17)$$
where $F_j$ is given by a $p_{j-1}\times p_j$ matrix of sections in 
$\cA(\Omega)$.

\subsection{3.3}{Analytic Sheaves and the Oka Theorem} 
Many properties of holomorphic functions which will be considered in
this book can be expressed in terms of sheaves.  Among them, analytic
sheaves play a central role.  The Oka theorem \cite{Oka 1950} asserting
the coherence of the sheaf of holomorphic functions can be seen as a
far-reaching deepening of the noetherian property seen in Sect.~1.
The theory of analytic sheaves could not be presented without it.

\claim{(3.18) Definition} Let $M$ be a $n$-dimensional complex analytic
manifold and let $\cO_M$ be the sheaf of germs of analytic functions on~$M$. 
An analytic sheaf over $M$ is by definition a sheaf $\cS$ of modules
over~$\cO_M$.
\endclaim

\claim{(3.19) Coherence theorem of Oka} The sheaf of rings
$\cO_M$ is coherent for any complex manifold~$M$.
\endclaim

Let $F_1,\ldots,F_q\in\cO(U)$. Since $\cO_{M,x}$ is Noetherian,  we already
know that every stalk $\cR(F_1,\ldots,F_q)_x\subset\cO_{M,x}^{\oplus q}$ is
finitely generated, but the important new fact expressed by the theorem
is that the sheaf of relations is locally finitely generated, namely
that the ``same'' generators can be chosen to generate each stalk in a
neighborhood of a given point.

\proof{Proof} By induction on $n=\dim_\bC M$. For $n=0$, the stalks 
$\cO_{M,x}$ are equal to $\bC$ and the result is trivial.
Assume now that $n\ge 1$ and that the result has already been proved 
in dimension $n-1$. Let $U$ be an open set of $M$ and
$F_1,\ldots,F_q\in\cO_M(U)$. To show that $\cR(F_1,\ldots,F_q)$ is
locally finitely generated, we may assume that
$U=\Delta=\Delta'\times\Delta_n$ is a polydisk in $\bC^n$ centered
at $x_0=0$~; after a change of coordinates and multiplication of 
$F_1,\ldots,F_q$ by invertible functions, we may also suppose that
$F_1,\ldots,F_q$ are Weierstrass polynomials in $z_n$ with coefficients
in $\cO(\Delta')$. We need a lemma.
\endproof

\claim{(3.20) Lemma} If $x=(x',x_n)\in\Delta$, the $\cO_{\Delta,x}$-module
$\cR(F_1,\ldots,F_q)_x$ is generated by those of its elements whose components
are germs of analytic polynomials in $\cO_{\Delta',x'}[z_n]$ with a degree in
$z_n$ at most equal to $\mu$, the maximum of the degrees of $F_1,\ldots,F_q$.
\endclaim

\proof{Proof} Assume for example that $F_q$ is of the maximum degree $\mu$.
By the Weierstrass preparation Th.~1.1 and Lemma~1.9
applied at $x$, we can write $F_{q,x}=f'f''$ where $f',f''\in\cO_{\Delta',x'}
[z_n]$, $f'$ is a Weierstrass polynomial in $z_n-x_n$ and $f''(x)\ne 0$.
Let $\mu'$ and $\mu''$ denote the degrees of $f'$ and $f''$ with respect to
$z_n$, so $\mu'+\mu''=\mu$. Now, take $(g^1,\ldots,g^q)\in \cR(F_1,\ldots,F_q)_x$. 
The Weierstrass division theorem gives
$$g^j=F_{q,x}t^j+r^j,~~~~j=1,\ldots,q-1,$$
where $t^j\in\cO_{\Delta,x}$ and $r^j\in\cO_{\Delta',x'}[z_n]$ is a polynomial
of degree $<\mu'$. For $j=q$, define $r^q=g^q+\sum_{1\le j\le q-1}
t^jF_{j,x}$. We can write
$$(g^1,\ldots,g^q)=\sum_{1\le j\le q}t^j(0,\ldots,F_q,\ldots,0,-F_j)_x+(r^1,\ldots,r^q)
\leqno(3.21)$$
where $F_q$ is in the $j$-th position in the $q$-tuples of the summation.
Since these
$q$-tuples are in $\cR(F_1,\ldots,F_q)_x$, we have $(r^1,\ldots,r^q)\in 
\cR(F_1,\ldots,F_q)_x$, thus
$$\sum_{1\le j\le q-1}F_{j,x}r^j+f'f''r^q=0.$$
As the sum is a polynomial in $z_n$ of degree $<\mu+\mu'$, it follows from
Lemma~1.9 that $f''r^q$ is a polynomial in $z_n$ of degree $<\mu$.
Now we have
$$(r^1,\ldots,r^q)=1/f''(f''r^1,\ldots,f''r^q)$$
where $f''r^j$ is of degree $<\mu'+\mu''=\mu$. In combination with (3.21)
this proves the lemma.\qed
\endproof

\proof{Proof of Theorem 3.19 {\rm (end).}} If $g=(g^1,\ldots,g^q)$
is one of the polynomials\break of 
$\cR(F_1,\ldots,F_q)_x$ described in Lemma~3.20,
we can write
$$g^j=\sum_{0\le k\le\mu}u^{jk}z_n^k,~~~~u^{jk}\in\cO_{\Delta',x'}.$$
The condition for $(g^1,\ldots,g^q)$ to belong to $\cR(F_1,\ldots,F_q)_x$ therefore
consists of $2\mu+1$ linear conditions for the germ $u=(u^{jk})$ with
coefficients in $\cO(\Delta')$. By the induction hypothesis, $\cO_{\Delta'}$
is coherent and Th.~3.16 shows that the corresponding modules of 
relations are generated over $\cO_{\Delta',x'}$, for $x'$ in a
neighborhood $\Omega'$ of $0$, by finitely many $(q\times\mu)$-tuples
$U_1,\ldots,U_N\in\cO(\Omega')^{q\mu}$. By Lemma~3.20, $\cR(F_1,\ldots,F_q)_x$
is generated at every point $x\in\Omega=\Omega'\times\Delta_n$ by the 
germs of the corresponding polynomials
$$G_l(z)=\Big(\sum_{0\le k\le\mu}U^{jk}_l(z')z_n^k\Big)_{1\le j\le q},
~~~~z\in\Omega,~~~1\le l\le N.\eqno\square$$
\endproof

\claim{(3.22) Strong Noetherian property} Let $\cF$ be a coherent
analytic sheaf on a complex manifold $M$ and let $\cF_1\subset\cF_2\subset
\ldots$ be an increasing sequence of coherent subsheaves of~$\cF$. Then
the sequence $(\cF_k)$ is stationary on every compact subset of~$M$.
\endclaim

\proof{Proof} Since $\cF$ is locally a quotient of a free module $\cO_M^{\oplus q}$,
we can pull back the sequence to $\cO_M^{\oplus q}$ and thus suppose $\cF=\cO_M$
(by easy reductions similar to those in the proof of Th.~3.14).
Suppose $M$ connected and $\cF_{k_0}\ne\{0\}$ for some index $k_0$
(otherwise, there is nothing to prove). By the analytic continuation
theorem, we easily see that $\cF_{k_0,x}\ne\{0\}$ for every $x\in M$.
We can thus find a non zero Weierstrass polynomial $P\in\cF_{k_0}(V)$,
${\rm deg}_{z_n}P(z',z_n)=\mu$, in a coordinate neighborhood
$V=\Delta'\times\Delta_n$ of any point $x\in M$. A division by $P$ 
shows that for $k\ge k_0$ and $x\in V$, all stalks $\cF_{k,x}$ are
generated by $P_x$ and by polynomials of degree $<\mu$ in $z_n$
with coefficients in $\cO_{\Delta',x'}$. Therefore, we can apply
induction on $n$ to the coherent $\cO_{\Delta'}$-module
$$\cF'=\cF\cap\big\{Q\in\cO_{\Delta'}[z_n]\,;\,{\rm deg}\,Q
\le\mu\big\}$$
and its increasing sequence of coherent subsheaves $\cF'_k=
\cF_k\cap\cF'$.\qed
\endproof

\section{4}{Complex Analytic Sets. Local Properties}
\subsection{4.1}{Definition. Irreducible Components}
A complex analytic set is a set which can be defined locally by finitely
many holomorphic equations; such a set has in general singular points,
because no assumption is made on the differentials of the equations.
We are interested both in the description of the singularities
and in the study of algebraic properties of holomorphic functions
on analytic sets. For a more detailed study than ours, we refer
to H.~Cartan's seminar \cite{Cartan 1950}, to the books of
\cite{Gunning-Rossi 1965}, \cite{Narasimhan 1966} or the recent book
by \cite{Grauert-Remmert 1984}.

\claim{(4.1) Definition} Let $M$ be a complex analytic manifold. A subset
$A\subset M$ is said to be an analytic subset of $M$ if $A$ is closed and if
for every point $x_0\in A$ there exist a neighborhood $U$ of $x_0$ and
holomorphic functions $g_1,\ldots,g_n$ in $\cO(U)$ such that
$$A\cap U=\{z\in U~;~g_1(z)=\ldots=g_N(z)=0\}.$$
Then $g_1,\ldots,g_N$ are said to be $($local$)$ equations of $A$ in $U$.
\endclaim

It is easy to see that a finite union or intersection of analytic sets is 
analytic: if $(g'_j)$, $(g''_k)$ are equations of $A'$, $A''$ in the 
open set $U$, then the family of all products $(g'_jg''_k)$ and the
family $(g'_j)\cup(g''_k)$ define equations of $A'\cup A''$ and $A'\cap A''$
respectively.

\claim{(4.2) Remark} \rm Assume that $M$ is connected. The analytic
continuation theorem shows that either $A=M$ or $A$ has no interior
point. In the latter case, each piece $A\cap U=g^{-1}(0)$ is the set
of points where the function
$\log|g|^2=\log(|g_1|^2+\cdots+|g_N|^2)\in\Psh(U)$ takes the value
$-\infty$, hence $A$ is pluripolar.  In particular $M\ssm A$ is
connected and every function $f\in\cO(M\ssm A)$ that is locally bounded
near $A$ can be extended to a function $\tilde f\in\cO(M)$.\qed
\endclaim

We focus now our attention on local properties of analytic sets. By
definition, a germ of set at a point $x\in M$ is an equivalence class
of elements in the power set $\cP(M)$, with $A\sim B$ if there is
an open neighborhood $V$ of $x$ such that $A\cap V=B\cap V$. The germ
of a subset $A\subset M$ at $x$ will be denoted by $(A,x)$. We most often
consider the case when $A\subset M$ is a analytic set in a neighborhood
$U$ of~$x$, and in this case we denote by $\cI_{A,x}$ the ideal of germs 
$f\in\cO_{M,x}$ which vanish on $(A,x)$. Conversely, if $\cJ=(g_1,\ldots,g_N)$ is
an ideal of $\cO_{M,x}$, we denote by $\big(V(\cJ),x\big)$ the germ at $x$ of 
the zero variety $V(\cJ)=\{z\in U~;~g_1(z)=\ldots=g_N(z)=0\}$, where 
$U$ is a neighborhood of $x$ such that $g_j\in\cO(U)$. It is easy to check
that the germ $(V(cJ),x)$ does not depend on the choice of generators.
Moreover, it is clear that
$$\leqalignno{
&\hbox{\rm for every ideal $\cJ$ in the ring $\cO_{M,x}$,}\qquad~~\,
\cI_{V(\cJ),x}\supset\cJ,&(4.3')\cr
&\hbox{\rm for every germ of analytic set $(A,x)$,}\qquad
\big(V(\cI_{A,x}),x\big)=(A,x).&(4.3'')\cr}$$

\claim{(4.4) Definition} A germ $(A,x)$ is said to be irreducible if it has
no decom\-po\-sition $(A,x)=(A_1,x)\cup(A_2,x)$ with analytic sets
\hbox{$(A_j,x)\ne(A,x)$, $j=1,2$}.
\endclaim

\claim{(4.5) Proposition} A germ $(A,x)$ is irreducible if and 
only if $\cI_{A,x}$ is a prime ideal of the ring $\cO_{M,x}$.
\endclaim

\proof{Proof} Let us recall that an ideal $\cJ$ is said to be {\it prime} if 
$fg\in\cJ$ implies $f\in\cJ$ or $g\in\cJ$. Assume that $(A,x)$ is
irreducible and that $fg\in\cI_{A,x}$. As we can write 
$(A,x)=(A_1,x)\cup(A_2,x)$ with $A_1=A\cap f^{-1}(0)$ and 
$A_2=A\cap g^{-1}(0)$, we must have for example
$(A_1,x)=(A,x)$~; thus $f\in\cI_{A,x}$ and $\cI_{A,x}$ is
prime. Conversely, if $(A,x)=(A_1,x)\cup(A_2,x)$ with $(A_j,x)\ne(A,x)$, 
there exist $f\in\cI_{A_1,x}$, $g\in\cI_{A_2,x}$ such that $f,g\notin
\cI_{A,x}$. However $fg\in\cI_{A,x}$, thus $\cI_{A,x}$ is not prime.\qed
\endproof

\claim{(4.6) Theorem} Every decreasing sequence of germs of analytic sets
$(A_k,x)$ is stationary.
\endclaim

\proof{Proof} In fact, the corresponding sequence of ideals $\cJ_k=\cI_{A_k,x}$
is increa\-sing, thus $\cJ_k=\cJ_{k_0}$ for $k\ge k_0$ large enough by
the Noetherian property of~$\cO_{M,x}$. Hence $(A_k,x)=\big(V(\cJ_k),x\big)$
is constant for $k\ge k_0$. This result has the following straightforward
consequence:\qed
\endproof

\claim{(4.7) Theorem} Every analytic germ $(A,x)$ has a finite decomposition
$$(A,x)=\bigcup_{1\le k\le N}(A_k,x)$$
where the germs $(A_j,x)$ are irreducible and $(A_j,x)\not\subset(A_k,x)$
for $j\ne k$. The decomposition is unique apart from the ordering.
\endclaim

\proof{Proof} If $(A,x)$ can be split in several components, we split repeatedly
each component as long as one of them is reducible. The process
must stop by Th.~4.6, whence the existence. For the uniqueness,
assume that $(A,x)=\bigcup(A'_l,x)$, $1\le l\le N'$, is another
decomposition. Since $(A_k,x)=\bigcup_l(A_k\cap A'_l,x)$, we must have
$(A_k,x)=(A_k\cap A'_l,x)$ for some $l=l(k)$, i.e.\ $(A_k,x)\subset
(A'_{l(k)},x)$, and likewise $(A'_{l(k)},x)\subset(A_j,x)$ for some~$j$. 
Hence $j=k$ and $(A'_{l(k)},x)=(A_k,x)$.\qed
\endproof

\subsection{4.2}{Local Structure of a Germ of Analytic Set}
We are going to describe the local structure of a germ, both from the 
holomorphic and topological points of view. By the above decomposition 
theo\-rem,  we may restrict ourselves to the case of irreducible germs
Let~$\cJ$ be a prime ideal of $\cO_n=\cO_{\bC^n,0}$ and let $A=V(\cJ)$
be its zero variety. We set $\cJ_k=\cJ\cap\bC\{z_1,\ldots,z_k\}$ for
each $k=0,1,\ldots,n$.

\claim{(4.8) Proposition} There exist an integer $d$, a basis
$(e_1,\ldots,e_n)$ of $\bC^n$ and associated coordinates $(z_1,\ldots,z_n)$
with the following properties: $\cJ_d=\{0\}$ and for every
integer $k=d+1,\ldots,n$ there is a Weierstrass polynomial $P_k\in\cJ_k$
of the form
$$P_k(z',z_k)=z_k^{s_k}+\sum_{1\le j\le s_k}a_{j,k}(z')\,z_k^{s_k-j},~~~~
a_{j,k}(z')\in\cO_{k-1},\leqno(4.9)$$
where $a_{j,k}(z')=O(|z'|^j)$. Moreover, the basis $(e_1,\ldots,e_n)$ can
be chosen arbitrarily close to any preassigned basis $(e^0_1,\ldots,e^0_n)$.
\endclaim

\proof{Proof} By induction on $n$. If $\cJ=\cJ_n=\{0\}$, then $d=n$ and
there is nothing to prove. Otherwise, select a non zero element 
$g_n\in\cJ$ and a vector $e_n$ such that $\bC\ni w\longmapsto g_n(we_n)$
has minimum vanishing order $s_n$. This choice excludes at most the
algebraic set $g_n^{(s_n)}(v)=0$, so $e_n$ can be taken arbitrarily
close to $e^0_n$. Let $(\tilde z_1,\ldots,\tilde z_{n-1},z_n)$ be the coordinates
associated to the basis $(e^0_1,\ldots,e^0_{n-1},e_n)$. After
multiplication by an invertible element, we may assume that $g_n$ is
a Weierstrass polynomial
$$P_n(\tilde z,z_n)=z_n^{s_n}+\sum_{1\le j\le s_n}a_{j,n}(\tilde z)\,z_n^{s_n-j},
~~~~a_{j,n}\in\cO_{n-1},$$
and $a_{j,n}(\tilde z)=O(|\tilde z|^j)$ by Remark~2.2. If
$\cJ_{n-1}=\cJ\cap\bC\{\tilde z\}=\{0\}$ then $d=n-1$ and the construction
is finished. Otherwise we apply the induction hypothesis to the
ideal $\cJ_{n-1}\subset\cO_{n-1}$ in order to find a new basis
$(e_1,\ldots,e_{n-1})$ of $\Vect(e^0_1,\ldots,e^0_{n-1})$,
associated coordinates $(z_1,\ldots,z_{n-1})$ and Weierstrass polynomials 
$P_k\in\cJ_k$, $d+1\le k\le n-1$, as stated in the lemma.\qed
\endproof

\claim{(4.10) Lemma} If $w\in\bC$ is a root of $w^d+a_1w^{d-1}+\cdots+a_d=0$,
$a_j\in\bC$, then $|w|\le 2\max|a_j|^{1/j}$.
\endclaim

\proof{Proof} Otherwise $|w|>2|a_j|^{1/j}$ for all $j=1,\ldots,d$ and the given
equation $-1=a_1/w+\cdots+a_d/w^d$ implies $1\le 2^{-1}+\cdots+2^{-d}$,
a contradiction.\qed
\endproof

\claim{(4.11) Corollary} Set $z'=(z_1,\ldots,z_d)$, $z''=(z_{d+1},\ldots,z_n)$, and let
$\Delta'$ in $\bC^d$, $\Delta''$ in $\bC^{n-d}$ be polydisks of center $0$
and radii $r',r''>0$. Then the germ $(A,0)$ is contained in a cone 
$|z''|\le C|z'|$, $C=\hbox{\it constant}$, and the restriction of the 
projection map $\bC^n\longrightarrow\bC^d$, $(z',z'')\longmapsto z'~:$
$$\pi:A\cap(\Delta'\times\Delta'')\longrightarrow\Delta'$$
is proper if $r''$ is small enough and $r'\le r''/C$.
\endclaim

\proof{Proof} The polynomials $P_k(z_1,\ldots,z_{k-1}\,;\,z_k)$ vanish on $(A,0)$. 
By Lemma~4.10 and (4.9), every point $z\in A$ sufficiently close to $0$
satisfies
$$|z_k|\le C_k(|z_1|+\cdots+|z_{k-1}|),~~~~d+1\le k\le n,$$
thus $|z''|\le C|z'|$ and the Corollary follows.\qed
\endproof

Since $\cJ_d=\{0\}$, we have an injective ring morphism 
$$\cO_d=\bC\{z_1,\ldots,z_d\}\lhra\cO_n/\cJ.\leqno(4.12)$$

\claim{(4.13) Proposition} $\cO_n/\cJ$ is a finite integral
extension of $\cO_d$.
\endclaim

\proof{Proof} Let $f\in\cO_n$. A division by $P_n$ yields $f=P_nq_n+R_n$ with 
a remainder $R_n\in\cO_{n-1}[z_n]$, $\deg_{z_n}R_n<s_n$. Further divisions
of the coefficients of $R_n$ by $P_{n-1}$, $P_{n-2}\,$ etc $\ldots$ yield
$$R_{k+1}=P_kq_k+R_k,~~~~R_k\in\cO_k[z_{k+1},\ldots,z_n],$$
where $\deg_{z_j}R_k<s_j$ for $j>k$. Hence
$$f=R_d+\sum_{d+1\le k\le n}P_kq_k=R_d~~~\hbox{\rm mod}~~(P_{d+1},\ldots,P_n)
\subset\cJ\leqno(4.14)$$
and $\cO_n/\cJ$ is finitely generated as an $\cO_d$-module by the
family of monomials $z_{d+1}^{\alpha_{d+1}}\ldots z_n^{\alpha_n}$ with
$\alpha_j<s_j$.\qed
\endproof

As $\cJ$ is prime, $\cO_n/\cJ$ is an entire ring. We denote by $\tilde f$
the class of any germ $f\in\cO_n$ in $\cO_n/\cJ$, by $\cM_A$ and $\cM_d$ 
the quotient fields of $\cO_n/\cJ$ and $\cO_d$ respectively. Then
$\cM_A=\cM_d[\tilde z_{d+1},\ldots,\tilde z_n]$ is a finite algebraic extension
of~$\cM_d$. Let $q=[\cM_A{:}\cM_d]$ be its degree and let
$\sigma_1,\ldots,\sigma_q$ be the embeddings of $\cM_A$ over $\cM_d$
in an algebraic closure $\ol\cM_A$. Let us recall that a factorial ring
is integrally closed in its quotient field (\cite{Lang~1965}, Chapter~IX). Hence
every element of $\cM_d$ which is integral over $\cO_d$ lies in fact in $\cO_d$.
By the primitive element theorem,
there exists a linear form $u(z'')=c_{d+1}z_{d+1}+\cdots+c_nz_n$, $c_k\in\bC$,
such that $\cM_A=\cM_d[\tilde u]$~; in fact, $u$ is of degree $q$ if and only if
$\sigma_1\tilde u,\ldots,\sigma_q\tilde u$ are all distinct, and this excludes at
most a finite number of vector subspaces in the space $\bC^{n-d}$ of
coefficients $(c_{d+1},\ldots,c_n)$.  As \hbox{$\tilde u\in\cO_n/\cJ$} is integral
over the integrally closed ring $\cO_d$, the unitary irreducible polynomial
$W_u$ of $\tilde u$ over $\cM_d$ has coefficients in $\cO_d$~:
$$W_u(z'\,;T)=T^q+\sum_{1\le j\le q}a_j(z_1,\ldots,z_d)\,T^{q-j},~~~~
a_j\in\cO_d.$$
$W_u$ must be a Weierstrass polynomial, otherwise there would exist a
facto\-rization $W_u=W'Q$ in $\cO_d[T]$ with a Weierstrass polynomial $W'$
of degree deg$\,W'<q=\deg\,\tilde u$ and $Q(0)\ne 0$, hence
$W'(\tilde u)=0$, a contradiction. In the same way, we see that
$\tilde z_{d+1},\ldots,\tilde z_n$ have irreducible equations $W_k(z'\,;\,\tilde z_k)=0$
where $W_k\in\cO_d[T]$ is a Weierstrass polynomial of 
degree $=\deg\,\tilde z_k\le q$, $d+1\le k\le n$.

\claim{(4.15) Lemma} Let $\delta(z')\in\cO_d$ be the discriminant of 
$W_u(z'\,;T)$. For every element $g$ of $\cM_A$ which is integral
over $\cO_d$ (or equivalently over $\cO_n/\cJ$)
we have $\delta g\in\cO_d[\tilde u]$.
\endclaim

\proof{Proof} We have $\delta(z')=\prod_{j<k}(\sigma_k\tilde u-\sigma_j\tilde u)^2\not
\equiv 0\,$, and $g\in\cM_A=\cM_d[\tilde u]$ can be written
$$g=\sum_{0\le j\le q-1}b_j\,\tilde u^j,~~~~b_j\in\cM_d,$$
where $b_0,\ldots,b_{d-1}$ are the solutions of the linear system
$\sigma_k g=\sum b_j(\sigma_k\tilde u)^j$~;
the determinant (of Van der Monde type) is $\delta^{1/2}$. It follows 
that $\delta b_j\in\cM_d$ are polynomials
in $\sigma_k g$ and $\sigma_k\tilde u$, thus $\delta b_j$ is integral over 
$\cO_d$. As $\cO_d$ is integrally closed, we must have $\delta b_j\in\cO_d$,
hence $\delta g\in\cO_d[\tilde u]$.\qed
\endproof

In particular, there exist unique polynomials $B_{d+1}$, $\ldots$, 
$B_n\in\cO_d[T]$ with deg$\,B_k\le q-1$, such that
$$\delta(z')z_k=B_k(z'\,;u(z''))~~~~\hbox{\rm (mod}~\cJ).\leqno(4.16)$$
Then we have
$$\delta(z')^qW_k\big(z'\,;B_k(z'\,;\,T)/\delta(z')\big)\in
\hbox{\rm ideal~~}W_u(z'\,;\,T)\,\cO_d[T]~;\leqno(4.17)$$
indeed, the left-hand side is a polynomial in $\cO_d[T]$ and admits
$T=\tilde u$ as a root in $\cO_n/\cJ$ since $B_k(z'\,;\,\tilde u)/\delta(z')=
\tilde z_k$ and $W_k(z'\,;\,\tilde z_k)=0$.

\claim{(4.18) Lemma} Consider the ideal 
$$\cG=\big(W_u(z'\,;u(z''))\,,\,\delta(z')z_k-B_k(z'\,;u(z''))\big)
\subset\cJ$$
and set $m=\max\{q,(n-d)(q-1)\}$. For every germ $f\in\cO_n$, 
there exists a unique polynomial $R\in\cO_d[T]$, $\deg_TR\le q-1$, 
such that
$$\delta(z')^mf(z)=R(z'\,;u(z''))~~~~\hbox{\rm (mod}~\cG).$$
Moreover $f\in\cJ$ implies $R=0$, hence $\delta^m\cJ\subset\cG$.
\endclaim

\proof{Proof} By (4.17) and a substitution of $z_k$, we find 
$\delta(z')^qW_k(z'\,;z_k)\in\cG$.  The analogue of 
formula (4.14) with $W_k$ in place of $P_k$ yields
$$f=R_d+\sum_{d+1\le k\le n}W_kq_k,~~~~R_d\in\cO_d[z_{d+1},\ldots,z_n],$$
with $\deg_{z_k}R_d<\deg\,W_k\le q$, thus $\delta^mf=\delta^mR_d$
mod $\cG$. We may therefore replace $f$ by $R_d$ and assume that
$f\in\cO_d[z_{d+1},\ldots,z_n]$ is a polynomial of total degree
$\le(n-d)(q-1)\le m$.  A substitution of $z_k$ by 
$B_k(z'\,;u(z''))/\delta(z')$ yields $G\in\cO_d[T]$ such that
$$\delta(z')^m f(z)=G(z'\,;u(z''))~~~~\hbox{\rm mod}~~\big(\delta(z')
z_k-B_k(z'\,;u(z''))\big).$$
Finally, a division $G=W_uQ+R$ gives the required polynomial
$R\in\cO_d[T]$. The last statement is clear: if $f\in\cJ$ satisfies 
$\delta^m(z')f(z)=R(z\,;u(z''))$ mod $\cG$ for $\deg_T R<q$, 
then $R(z'\,;\tilde u)=0$, and as $\tilde u\in\cO_n/\cJ$ is of 
degree $q$, we must have $R=0$. The uniqueness of $R$ is proved 
similarly.\qed
\endproof

\claim{(4.19) Local parametrization theorem} Let $\cJ$ be a prime
ideal of $\cO_n$ and let $A=V(\cJ)$. Assume that the coordinates
$$(z'\,;z'')=(z_1,\ldots,z_d\,;z_{d+1},\ldots,z_n)$$
are chosen as above. Then the ring $\cO_n/\cJ$ is a finite integral
extension of~$\cO_d\,$; let $q$ be the degree of the extension and let
$\delta(z')\in\cO_d$ be the discriminant of the irreducible polynomial
of a primitive element $u(z'')=\sum_{k>d}c_kz_k$.  If
$\Delta',\Delta''$ are polydisks of sufficiently small  radii $r',r''$
and if $r'\le r''/C$ with $C$ large, the projection map 
$\pi:A\cap(\Delta'\times\Delta'')\longrightarrow\Delta'$ is a ramified  covering
with $q$ sheets, whose ramification locus is contained in
$S=\{z'\in\Delta';\delta(z')=0\}$. This means that:
\smallskip
\item{\rm a)} the open subset $A_S=A\cap\big((\Delta'\ssm S)
\times\Delta''\big)$ is a smooth $d$-dimensional manifold,
dense in $A\cap(\Delta'\times\Delta'')~;$
\smallskip
\item{\rm b)} $\pi:A_S\longrightarrow\Delta'\ssm S$ is a covering~$;$
\smallskip
\item{\rm c)} $\,$the fibers $\pi^{-1}(z')$ have exactly $q$ elements
if $z'\notin S$ and at most $q$ if $z'\in S.$
\smallskip
\noindent Moreover, $A_S$ is a connected covering of $\Delta'\ssm S$,
and $A\cap(\Delta'\times\Delta'')$ is contained in a cone $|z''|\le C|z'|$~
$($see Fig. 1$)$.
\endclaim

\vbox{%
\input epsfiles/fig_2_1.tex
\vskip6mm
\centerline{{\bf Fig.~II-1} Ramified covering from $A$ to 
$\Delta'\subset\bC^p$.}}
\vskip5mm

\proof{Proof} After a linear change in the coordinates $z_{d+1},\ldots,z_n$, we may
assume $u(z'')=z_{d+1}$, so $W_u=W_{d+1}$ and $B_{d+1}(z'\,;T)=\delta(z')T$.
By Lemma~4.18, we have 
$$\cG=\big(W_{d+1}(z',z_{d+1})\,,\,\delta(z')z_k-B_k(z',z_{d+1})\big)_{k\ge d+2}
\subset\cJ,~~~~\delta^m\cJ\subset\cG.$$
We can thus find a polydisk $\Delta=\Delta'\times\Delta''$ of 
sufficiently small radii $r',r''$ such that $V(\cJ)\subset V(\cG)\subset
V(\delta^m\cJ)$ in~$\Delta$. As $V(\cJ)=A$ and $V(\delta)\cap\Delta=S\times
\Delta''$, this implies
$$A\cap\Delta\subset V(\cG)\cap\Delta\subset(A\cap\Delta)\cup
(S\times\Delta'').$$
In particular, the set $A_S=A\cap\big((\Delta'\ssm S)\times\Delta''\big)$
lying above $\Delta'\ssm S$ coincides with $V(\cG)\cap\big((\Delta'\ssm
S)\times\Delta''\big)$, which is the set of points $z\in\Delta$ parametrized
by the equations
$$\cases{
\delta(z')\ne 0,~~~W_{d+1}(z',z_{d+1})=0,\cr
z_k=B_k(z',z_{d+1})/\delta(z'),~~d+2\le k\le n.\cr}\leqno(4.20)$$
As $\delta(z')$ is the resultant of $W_{d+1}$ and 
$\partial W_{d+1}/\partial T$, we have
$$\partial W_{d+1}/\partial T(z',z_{d+1})\ne 0~~~~\hbox{\rm on}~~A_S.$$ 
The implicit function theorem shows that $z_{d+1}$ is locally a 
holomorphic function of $z'$ on $A_S$, and the same is true 
for $z_k=B_k(z',z_{d+1})/\delta(z')$, $k\ge d+2$. 
Hence $A_S$ is a smooth manifold, and for $r'\le r''/C$ small, 
the projection map $\pi:A_S\longrightarrow\Delta'\ssm S$ is a proper local 
diffeomorphism; by (4.20) the fibers $\pi^{-1}(z')$ have at most $q$ 
points corresponding to some of the $q$ roots $w$ of $W_{d+1}(z'\,;w)=0$. 
Since $\Delta'\ssm S$ is connected (Remark~4.2), either $A_S=\emptyset$
or the map $\pi$ is a covering of constant sheet number $q'\le q$.
However, if $w$ is a root of $W_{d+1}(z',w)=0$ with $z'\in\Delta'\ssm S$
and if we set $z_{d+1}=w$, $z_k=B_k(z',w)/\delta(z')$, $k\ge d+2$,
relation $(4.17)$ shows that $W_k(z',z_k)=0$, in particular
$|z_k|=O(|z'|^{1/q})$ by Lemma~4.10. For $z'$ small enough, 
the $q$ points $z=(z',z'')$ defined in this way lie in $\Delta$, 
thus $q'=q$. Property b) and the first parts of a) and c) follow. 
Now, we need the following lemma.
\endproof

\claim{(4.21) Lemma} If $\cJ\subset\cO_n$ is prime and $A=V(\cJ)$, 
then $\cI_{A,0}=\cJ$.
\endclaim

\proof It is obvious that $\cI_{A,0}\supset\cJ$. Now, for any
$f\in\cI_{A,0}$, Prop.~4.13 implies that $\tilde f$ satisfies
in $\cO_n/\cI$ an irreducible equation
$$f^r+b_1(z')\,f^{r-1}+\cdots+b_r(z')=0~~~~\hbox{\rm (mod}~\cJ).$$
Then $b_r(z')$ vanishes on $(A,0)$ and the first part of c) gives
$b_r=0$ on $\Delta'\ssm S$. Hence $\tilde b_r=0$ and
the irreducibility of the equation of~$\tilde f$ implies $r=1$, so
\hbox{$f\in\cJ$,} as desired.\qed
\endproof

\proof{Proof of Theorem 4.19 (end).}
It only remains to prove that $A_S$ is connected and dense in $A\cap\Delta$
and that the fibers $\pi^{-1}(z')$, $z'\in S$, 
have at most $q$ elements. Let $A_{S,1},\ldots,A_{S,N}$ be the connected components of
$A_S$. Then $\pi:A_{S,j}\longrightarrow\Delta'\ssm S$ is a covering with $q_j$ sheets, 
$q_1+\cdots+q_N=q$. For every point $\zeta'\in\Delta'\ssm S$, there 
exists a neighborhood $U$ of $\zeta'$ such that $A_{S,j}\cap\pi^{-1}(U)$ is a
disjoint union of graphs $z''=g_{j,k}(z')$ of analytic functions $g_{j,k}\in
\cO(U)$, $1\le k\le q_j$. If $\lambda(z'')$ is an arbitrary linear form in 
$z_{d+1},\ldots,z_n$ and $z'\in\Delta'\ssm S$, we set
$$P_{\lambda,j}(z'\,;T)=\prod_{\{z''\,;\,(z',z'')\in A_{S,j}\}}\big(T-\lambda
(z'')\big)=\prod_{1\le k\le k_j}\big(T-\lambda\circ g_{j,k}(z')\big).$$
This defines a polynomial in $T$ with bounded analytic coefficients on
$\Delta'\ssm S$. These coefficients have analytic extensions to
$\Delta'$ (Remark~4.2), thus $P_{\lambda,j}\in\cO(\Delta')[T]$. By
construction, $P_{\lambda,j}\big(z'\,;\lambda(z'')\big)$ vanishes identically
on $A_{S,j}$. Set
$$P_\lambda=\prod_{1\le j\le N}P_{\lambda,j},~~~~
f(z)=\delta(z')\,P_\lambda\big(z'\,;\lambda(z'')\big)~;$$
$f$ vanishes on $A_{S,1}\cup\ldots\cup A_{S,N}\cup(S\times\Delta'')\supset 
A\cap\Delta$.
Lemma~4.21 shows that $\cI_{A,0}$ is prime; as $\delta\notin\cI_{A,0}$, 
we get $P_{\lambda,j}\big(z'\,;\lambda(z'')\big)\in\cI_{A,0}$
for some~$j$. This is a contradiction if $N\ge 2$ and if $\lambda$ is chosen 
in such a way that $\lambda$ separates the $q$ points $z''_\nu$ in each fiber
$\pi^{-1}(z'_\nu)$, for a sequence $z'_\nu\to 0$ in $\Delta'\ssm S$.
Hence $N=1$, $A_S$ is connected, and for every $\lambda\in(\bC^{n-d})^\star$ 
we have $P_\lambda\big(z',\lambda(z'')\in\cI_{(A,0)}$. By construction
$P_\lambda\big(z',\lambda(z'')\big)$ vanishes on $A_S$, so it vanishes 
on $\ol A_S$~; hence, for every $z'\in S$, the fiber 
$\ol A_S\cap\pi^{-1}(z')$ has at most $q$ elements, 
otherwise selecting $\lambda$ which separates $q+1$ of these points would
yield $q+1$ roots $\lambda(z'')$ of $P_\lambda(z'\,;T)$, a
contradiction. Assume now that
$A_S$ is not dense in $A\cap\Delta$ for arbitrarily small polydisks $\Delta$.
Then there exists a sequence $A\ni z_\nu=(z'_\nu,z''_\nu)\to 0$ such that
$z'_\nu\in S$ and $z''_\nu$ is not in $F_\nu:=\hbox{\rm pr}''\big(\ol A_S\cap
\pi^{-1}(z'_\nu)\big)$. The continuity of the roots of the polynomial
$P_\lambda(z'\,;T)$ as $\Delta'\ssm S\ni z'\to z'_\nu$ implies that 
the set of roots of $P_\lambda(z'_\nu\,;T)$ is $\lambda(F_\nu)$. Select 
$\lambda$ such that $\lambda(z''_\nu)\notin\lambda(F_\nu)$ for all $\nu$. 
Then $P_\lambda\big(z'_\nu\,;\lambda(z''_\nu)\big)\ne 0$ for every $\nu$ and
$P_\lambda\big(z'\,;\lambda(z'')\big)\notin\cI_{A,0}$, a 
contradiction.\qed
\endproof

At this point, it should be observed that many of the above statements
completely fail in the case of real analytic sets. In $\bR^2$, for example,
the prime ideal $\cJ=(x^5+y^4)$ defines an irreducible germ 
of curve $(A,0)$ and there is an injective integral extension of rings
$\bR\{x\}\lhra\bR\{x,y\}/\cJ$ of degree 4; however, the projection of $(A,0)$
on the first factor, $(x,y)\mapsto x$, has not a constant sheet number 
near $0$, and this number is not related to the degree of the extension.
Also, the prime ideal $\cJ=(x^2+y^2)$ has an associated variety $V(\cJ)$
reduced to $\{0\}$, hence $\cI_{A,0}=(x,y)$ is strictly larger than $\cJ$,
in contrast with Lemma~4.21.

Let us return to the complex situation, which is much better behaved.
The result obtained in Lemma~4.21 can then be extended to non prime
ideals and we get the following important result:

\claim{(4.22) Hilbert's Nullstellensatz} For every ideal 
$\cJ\subset\cO_n$
$$\cI_{V(\cJ),0}=\sqrt{\cJ},$$
where $\sqrt{\cJ}$ is the radical of $\cJ$, i.e.\ the set of germs $f\in\cO_n$ 
such that some power $f^k$ lies in $\cJ$.
\endclaim

\proof{Proof} Set $B=V(\cJ)$. If $f^k\in\cJ$, then $f^k$ vanishes on $(B,0)$
and $f\in\cI_{B,0}$. Thus $\sqrt{\cJ}\subset\cI_{B,0}$. Conversely, 
it is well known that $\sqrt{\cJ}$ is the intersection
of all prime ideals $\cP\supset\cJ$ (\cite{Lang~1965}, Chapter~VI).
For such an ideal
$(B,0)=\big(V(\cJ),0)\supset\big(V(\cP),0\big)$, thus
$\cI_{B,0}\subset\cI_{V(\cP),0}=\cP$ in view of Lemma~4.21. Therefore
$\cI_{B,0}\subset\bigcap_{\cP\supset\cJ}\cP=\sqrt{\cJ}$ and the
Theorem is proved.\qed
\endproof

In other words, if a germ $(B,0)$ is defined by an arbitrary 
ideal $\cJ\subset\cO_n$ and if $f\in\cO_n$ vanishes on $(B,0)$,
then some power $f^k$ lies in $\cJ$.

\subsection{4.3}{Regular and Singular Points. Dimension}
The above powerful results enable us to investigate the structure of
singularities of an analytic set. We first give a few definitions.

\claim{(4.23) Definition} Let $A\subset M$ be an analytic set and $x\in A$.
We say that $x\in A$ is a regular point of $A$ if $A\cap\Omega$ is a 
$\bC$-analytic submanifold of $\Omega$
for some neighborhood $\Omega$ of $x$. Otherwise $x$ is said to 
be singular. The corresponding subsets of $A$ will be denoted respectively
$A_\reg$ and $A_\sing$.
\endclaim

It is clear from the definition that $A_\reg$ is an open subset of $A$
(thus $A_\sing$ is closed), and that the connected components of
$A_\reg$ are $\bC$-analytic submanifolds of $M$ (non necessarily
closed).

\claim{(4.24) Proposition} If $(A,x)$ is irreducible, there exist 
arbitrarily small neighborhoods $\Omega$ of $x$ such that $A_\reg\cap\Omega$
is dense and connected in $A\cap\Omega$.
\endclaim

\proof{Proof} Take $\Omega=\Delta$ as in Th.~4.19. Then
$A_S\subset A_\reg\cap\Omega\subset A\cap\Omega$, where
$A_S$ is connected and dense in $A\cap\Omega$~; hence $A_\reg\cap\Omega$
has the same properties.\qed
\endproof

\claim{(4.25) Definition} The dimension of an irreducible germ of analytic
set $(A,x)$ is defined by $\dim(A,x)=\dim(A_\reg,x)$.
If $(A,x)$ has several irreducible components $(A_l,x)$, we set
$$\dim (A,x)=\max\{\dim(A_l,x)\},~~~~\codim(A,x)=n-\dim(A,x).$$
\endclaim

\claim{(4.26) Proposition} Let $(B,x)\subset(A,x)$ be germs of analytic sets.
If $(A,x)$ is irreducible and $(B,x)\ne(A,x)$, then
$\dim(B,x)<\dim(A,x)$ and $B\cap\Omega$ has empty interior in $A\cap\Omega$
for all sufficiently small neighborhoods $\Omega$ of $x$.
\endclaim

\proof{Proof} We may assume $x=0$, $(A,0)\subset(\bC^n,0)$ and $(B,0)$
irreducible.  Then $\cI_{A,0} \subset\cI_{B,0}$ are prime ideals.  When
we choose suitable coordinates for the ramified coverings, we may at
each step select vectors $e_n,e_{n-1},\ldots$ that work simultaneously
for $A$ and $B$.  If $\dim B=\dim A$, the process stops for both at the
same time, i.e.\  we get ramified coverings
$$\pi:A\cap(\Delta'\times\Delta'')\longrightarrow\Delta',~~~~
  \pi:B\cap(\Delta'\times\Delta'')\longrightarrow\Delta'$$
with ramification loci $S_A,S_B$.  Then $B\cap\big((\Delta'\ssm
(S_A\cup S_B))\times\Delta''\big)$ is an open subset of the manifold
$A_S=A\cap\big((\Delta'\ssm S_A)\times\Delta''\big)$, therefore
$B\cap A_S$ is an analytic subset of $A_S$ with non empty interior. 
The same conclusion would hold if $B\cap\Delta$ had non empty interior in
$A\cap\Delta$. As $A_S$ is connected, we get $B\cap A_S=A_S$, and as
$B\cap\Delta$ is closed in $\Delta$ we infer $B\cap\Delta\supset\ol
A_S=A\cap\Delta$, hence $(B,0)=(A,0)$, in contradiction with the
hypothesis.\qed
\endproof

\claim{(4.27) Example: parametrization of curves} {\rm Suppose that $(A,0)$
is an irreducible germ of curve ($\dim(A,0)=1$). If the disk
$\Delta'\subset\bC$ is chosen so small that $S=\{0\}$, then $A_S$ is a 
connected covering of $\Delta'\ssm\{0\}$ with $q$ sheets. Hence,
there exists a covering isomorphism between $\pi$ and the standard covering
$$\bC\supset\Delta(r)\ssm\{0\}\longrightarrow \Delta(r^q)\ssm\{0\},~~~~
t\longmapsto t^q,~~~~r^q=\hbox{\rm radius of~~}\Delta',$$
i.e.\ a map $\gamma:\Delta(r)\ssm\{0\}\longrightarrow A_S$ such that 
$\pi\circ\gamma(t)=t^q$. This map extends into a bijective holomorphic map
$\gamma:\Delta(r)\longrightarrow A\cap\Delta$ with $\gamma(0)=0$. 
This means that every irreducible germ of curve can be parametrized by a 
bijective holomorphic map defined on a disk in $\bC$
(see also Exercise~10.8).}
\endclaim

\subsection{4.4}{Coherence of Ideal Sheaves}
Let $A$ be an analytic set in a complex manifold $M$. The
{\it sheaf of ideals} $\cI_A$ is the subsheaf of $\cO_M$ consisting
of germs of holomorphic functions on $M$ which vanish on $A$. Its
stalks are the ideals $\cI_{A,x}$ already considered; note that 
$\cI_{A,x}=\cO_{M,x}$ if $x\notin A$.
If $x\in A$, we let $\cO_{A,x}$ be the ring of germs of functions on $(A,x)$ 
which can be extended as germs of holomorphic functions on $(M,x)$. 
By definition, there is a surjective morphism $\cO_{M,x}\longrightarrow\cO_{A,x}$
whose kernel is $\cI_{A,x}$, thus 
$$\cO_{A,x}=\cO_{M,x}/\cI_{A,x},~~~~\forall x\in A,\leqno(4.28)$$
i.e.\ $\cO_A=(\cO_M/\cI_A)_{\restriction A}$. Since $\cI_{A,x}=\cO_{M,x}$
for $x\notin A$, the quotient sheaf $\cO_M/\cI_A$ is zero on $M\ssm A$.

\claim{(4.29) Theorem {\rm (\cite{Cartan 1950})}} For any analytic set
$A\subset M$, the sheaf of ideals $\cI_A$ is a coherent analytic sheaf.
\endclaim

\proof{Proof} It is sufficient to prove the result when $A$ is an analytic
subset in a neighborhood of $0$ in $\bC^n$. If $(A,0)$ is not irreducible,
there exists a neighborhood $\Omega$ such that $A\cap\Omega=A_1\cup\ldots\cup
A_N$ where $A_k$ are analytic sets in $\Omega$ and $(A_k,0)$ is irreducible.
We have $\cI_{A\cap\Omega}=\bigcap\cI_{A_k}$, so by Cor.~3.15 we may
assume that $(A,0)$ is irreducible. Then we can choose coordinates \hbox{$z'$,
$z''$}, polydisks $\Delta',\Delta''$ and a primitive element $u(z'')=c_{d+1}
z_{d+1}+\cdots+c_nz_n$ such that Th.~4.19 is valid. Since $\delta(z')=
\prod_{j<k}(\sigma_k\tilde u-\sigma_j\tilde u)^2$, we see that $\delta(z')$
is in fact a polynomial in the $c_j\,$'s with coefficients in $\cO_d$. The same
is true for the coefficients of the polynomials $W_u(z'\,;T)$ and $B_k(z'\,;T)$
which can be expressed in terms of the elementary symmetric functions of the 
$\sigma_k\tilde u\,$'s. We suppose that $\Delta'$ is chosen small enough in 
order that all coefficients of these $\cO_d[c_{d+1},\ldots,c_n]$ polynomials
are in $\cO(\Delta')$. Let $\delta_\alpha\in\cO(\Delta')$ be some non 
zero coefficient appearing in $\delta^m=\sum\delta_\alpha c^\alpha$. 
Also, let $G_1,\ldots,G_N\in\cO(\Delta')[z'']$ be the 
coefficients of all monomials $c^\alpha$ appearing in the expansion of 
the functions $W_u(z'\,;u(z''))$ or $\delta(z')z_k-B_k(z'\,;u(z''))$.
Clearly, $G_1,\ldots,G_N$ vanish on $A\cap\Delta$. We contend that
$$\cI_{A,x}=\big\{f\in\cO_{M,x}~;~\delta_\alpha f\in(G_{1,x},\ldots,G_{N,x})\big\}.
\leqno(4.30)$$
This implies that the sheaf $\cI_A$ is the projection on the first factor of 
the sheaf of relations $\cR(\delta_\alpha,G_1,\ldots,G_N)\subset\cO_\Delta^{N+1}$,
which is coherent by the Oka theorem; Theorem~4.29 then follows.

We first prove that the inclusion $\cI_{A,x}\supset\{\ldots\}$ holds
in (4.30). In fact,
if $\delta_\alpha f\in(G_{1,x},\ldots,G_{N,x})$, then $f$ vanishes on
$A\ssm\{\delta_\alpha=0\}$ in some neighborhood of $x$. Since
$(A\cap\Delta)\ssm\{\delta_\alpha=0\}$ is dense in $A\cap\Delta$,
we conclude that $f\in\cI_{A,x}$.

To prove the other inclusion $\cI_{A,x}\subset\{\ldots\}$, we repeat
the proof of Lemma~4.18 with a few modifications. Let $x\in\Delta$
be a given point. At $x$, the irreducible polynomials $W_u(z'\,;T)$
and $W_k(z'\,;T)$ of $\tilde u$ and $\tilde z_k$ in $\cO_{M,0}/\cI_{A,0}$
split into
$$\eqalign{
W_u(z'\,;T)&=W_{u,x}\big(z'\,;T-u(x'')\big)\,Q_{u,x}\big(z'\,;T-u(x'')\big),\cr
W_k(z'\,;T)&=W_{k,x}(z'\,;T-x_k)\,Q_{k,x}(z'\,;T-x_k),\cr}$$
where $W_{u,x}(z'\,;T)$ and $W_{k,x}(z'\,;T)$ are Weierstrass
polynomials in $T$ and $Q_{u,x}(x',0)\ne 0$, $Q_{k,x}(x',0)\ne 0$.
For all $z'\in\Delta'$, the roots of $W_u(z'\,;T)$ are the values
$u(z'')$ at all points $z\in A\cap\pi^{-1}(z')$. As $A$ is closed, any 
point $z\in A\cap\pi^{-1}(z')$ with $z'$ near $x'$ has to be in a small 
neighborhood of one of the points $y\in A\cap\pi^{-1}(x')$. Choose 
$c_{d+1},\ldots,c_n$ such that the linear form $u(z'')$ separates all
points in the fiber $A\cap\pi^{-1}(x')$. Then, for a root $u(z'')$ of
$W_{u,x}\big(z'\,;T-u(x'')\big)$, the point $z$ must be in a neighborhood of 
$y=x$, otherwise $u(z'')$ would be near $u(y'')\ne u(x'')$ and
the Weierstrass polynomial $W_{u,x}(z'\,;T)$ would have a root away 
from $0$, in contradiction with (4.10). Conversely, if
$z\in A\cap\pi^{-1}(z')$ is near $x$, then
$Q_{u,x}\big(z'\,;u(z'')-u(x'')\big)\ne 0$ and $u(z'')$ is a root of
$W_{u,x}\big(z'\,;T-u(x'')\big)$. From this, we infer that every 
polynomial $P(z'\,;T)\in\cO_{\Delta',x'}[T]$ such that
$P\big(z'\,;u(z'')\big)=0$ on $(A,x)$ is a multiple of
$W_{u,x}\big(z'\,;T-u(x'')\big)$, because the roots of the latter
polynomial are simple for $z'$ in the dense set $(\Delta'\ssm S,x)$.
In particular $\deg\,P<\deg\,W_{u,x}$ implies $P=0$ and
$$\delta(z')^qW_{k,x}\big(z'\,;B_k(z'\,;u(z''))/\delta(z')-x_k\big)$$
is a multiple of $W_{u,x}\big(z'\,;T-u(x'')\big)$. If we replace
$W_u$, $W_k$ by $W_{u,x}$, $W_{k,x}$ respectively, 
the proof of Lemma~4.18 shows that for every $f\in\cO_{M,x}$ there is a
polynomial $R\in\cO_{\Delta',x'}[T]$ of degree $\deg\,R<\deg\,W_{u,x}$ 
such that
$$\eqalign{
&\delta(z')^mf(z)=R\big(z'\,;u(z'')\big)~~~~\hbox{\rm modulo the ideal}\cr
&\big(~W_{u,x}\big(z'\,;u(z'')-u(x'')\big),~\delta(z')z_k-
B_k\big(z'\,;u(z'')\big)~\big),\cr}$$
and $f\in\cI_{A,x}$ implies $R=0$. Since $W_{u,x}$ differs from $W_u$
only by an invertible element in $\cO_{M,x}$, we conclude that
$$\Big(\sum\delta_\alpha c^\alpha\Big)\cI_{A,x}=\delta^m\cI_{A,x}\subset
(G_{1,x},\ldots,G_{N,x}).$$
This is true for a dense open set of coefficients $c_{d+1},\ldots,c_n$,
therefore by expressing the coefficients $\delta_\alpha$ through
interpolation of $\sum \delta_\alpha c^\alpha$ at suitable points $c$
we infer
$$\delta_\alpha\cI_{A,x}\subset
(G_{1,x},\ldots,G_{N,x})~~~~\hbox{\rm for all~~}\alpha.\eqno\square$$
\endproof

\claim{(4.31) Theorem} $A_\sing$ is an analytic subset of $A$.
\endclaim

\proof{Proof} The statement is local. Assume first that $(A,0)$ is an 
irreducible germ in $\bC^n$. Let $g_1,\ldots,g_N$ be generators of the sheaf
$\cI_A$ on a neighborhood $\Omega$ of $0$. Set $d=\dim A$. In a neighborhood
of every point $x\in A_\reg\cap\Omega$, $A$ can be defined by holomorphic
equations $u_1(z)=\ldots=u_{n-d}(z)=0$ such that $du_1,\ldots,du_{n-d}$ are 
linearly independant. As $u_1,\ldots,u_{n-d}$ are generated by $g_1,\ldots,g_N$,
one can extract a subfamily $g_{j_1},\ldots,g_{j_{n-d}}$ that has at least one
non zero Jacobian determinant of rank $n-d$ at $x$. Therefore 
$A_\sing\cap\Omega$ is defined by the equations
$$\det\Big({\partial g_j\over\partial z_k}\Big)_{\scriptstyle j\in J
\atop\scriptstyle k\in K}=0,~~~~
J\subset\{1,\ldots,N\},~~K\subset\{1,\ldots,n\},~~|J|=|K|=n-d.$$
Assume now that $(A,0)=\bigcup(A_l,0)$ with $(A_l,0)$ irreducible. The germ 
of an analytic set at a regular point is irreducible, thus every point which 
belongs simultaneously to at least two components is singular. Hence
$$(A_\sing,0)=\bigcup(A_{l,\sing},0)\cup
\bigcup_{k\ne l}(A_k\cap A_l,0),$$
and $A_\sing$ is analytic.\qed
\endproof

Now, we give a characterization of regular points in terms of a simple
algebraic pro\-perty of the ring $\cO_{A,x}$.

\claim{(4.32) Proposition} Let $(A,x)$ be a germ of analytic set
of dimension $d$ and let $\gm_{A,x}\subset\cO_{A,x}$ be the maximal ideal
of functions that vanish at $x$. Then $\gm_{A,x}$ cannot have
less than $d$ generators and $\gm_{A,x}$ has $d$ generators if and
only if $x$ is a regular point.
\endclaim

\proof{Proof} If $A\subset\bC^n$ is a $d$-dimensional submanifold in a 
neighborhood of $x$, there are local coordinates centered at $x$
such that $A$ is given by the equations $z_{d+1}=\ldots=z_n$ near $z=0$.
Then $\cO_{A,x}\simeq\cO_d$ and $\gm_{A,x}$ is generated by $z_1,\ldots,z_d$.
Conversely, assume that $\gm_{A,x}$ has $s$ generators $g_1(z),\ldots,g_s(z)$
in $\cO_{A,x}=\cO_{\bC^n,x}/\cI_{A,x}$. Letting $x=0$ for simplicity, 
we can write
$$z_j=\sum_{1\le k\le s}u_{jk}(z)g_k(z)+f_j(z),~~~~u_{jk}\in\cO_n,~~
f_j\in\cI_{A,0},~~1\le j\le n.$$
Then we find $dz_j=\sum c_{jk}(0)dg_k(0)+df_j(0)$, so that the rank of 
the system of differentials $\big(df_j(0)\big)_{1\le j\le n}$ is at least
equal to $n-s$. Assume for example that $df_1(0),\ldots,df_{n-s}(0)$ are
linearly independent. By the implicit function theorem, the equations
$f_1(z)=\ldots=f_{n-s}(z)=0$ define a germ of sub\-manifold of dimension $s$
containing $(A,0)$, thus $s\ge d$ and $(A,0)$ equals this
submanifold if $s=d$.\qed
\endproof

\claim{(4.33) Corollary} Let $A\subset M$ be an analytic set of pure dimension
$d$ and let $B\subset A$ be an analytic subset of codimension $\ge p$ in~$A$.
Then, as an $\cO_{A,x}$-module, the ideal $\cI_{B,x}$ cannot be 
generated by less than $p$ generators at any point $x\in B$, 
and by less than $p+1$ generators at any point 
$x\in B_\reg\cap A_\sing$.
\endclaim

\proof{Proof} Suppose that $\cI_{B,x}$ admits $s$-generators
$(g_1,\ldots,g_s)$ at $x$. By coherence of $\cI_B$ these
germs also generate $\cI_B$ in a neighborhood of $x$,
so we may assume that $x$ is a regular point
of~$B$. Then there are local coordinates $(z_1,\ldots,z_n)$ on $M$
centered at $x$ such that $(B,x)$ is defined by $z_{k+1}=\ldots=z_n=0$,
where $k=\dim(B,x)$. Then the maximal ideal
$\gm_{B,x}=\gm_{A,x}/\cI_{B,x}$ is generated by 
$z_1,\ldots,z_k$, so that $\gm_{A,x}$ is generated by
$(z_1,\ldots,z_k,g_1,\ldots,g_s)$. By Prop.~4.32, we get
$k+s\ge d$, thus $s\ge d-k\ge p$, and we have strict inequalities 
when $x\in A_\sing$.\qed
\endproof

\section{5}{Complex Spaces}
Much in the same way a manifold is constructed by piecing together
open patches isomorphic to open sets in a vector space, a complex
space is obtained by gluing together open patches isomorphic to
analytic subsets. The general concept of analytic morphism 
(or holomorphic map between analytic sets) is first needed.

\subsection{5.1}{Morphisms and Comorphisms}
Let $A\subset\Omega\subset\bC^n$ and $B\subset\Omega'\subset\bC^p$ be analytic
sets. A morphism from $A$ to $B$ is by definition a map $F:A\longrightarrow B$ such that
for every $x\in A$ there is a neighborhood $U$ of $x$ and a holomorphic map
$\tilde F:U\longrightarrow\bC^p$ such that $\tilde F_{\restriction A\cap U}=F_{\restriction 
A\cap U}$. Equivalently, such a morphism can be defined as a continuous map 
$F:A\longrightarrow B$ such that for all $x\in A$ and $g\in\cO_{B,F(x)}$ we have
$g\circ F\in\cO_{A,x}$. The induced ring morphism
$$F^\star_x~:~~\cO_{B,F(x)}\ni g\longmapsto g\circ F\in\cO_{A,x}
\leqno(5.1)$$
is called the {\it comorphism} of $F$ at point $x$.

\subsection{5.1}{Definition of Complex Spaces}
\claim{(5.2) Definition} A complex space $X$ is a locally compact Hausdorff
space, countable at infinity, together with a sheaf $\cO_X$ of continuous
functions on~$X$, such that there exists an open covering $(U_\lambda)$ of 
$X$ and for each $\lambda$ a homeomorphism $F_\lambda:U_\lambda\longrightarrow
A_\lambda$ onto an analytic set $A_\lambda\subset\Omega_\lambda\subset
\bC^{n_\lambda}$ such that the comorphism $F^\star_\lambda:\cO_{A_\lambda}\longrightarrow
\cO_{X\,\restriction U_\lambda}$ is an isomorphism of sheaves of~rings. 
$\cO_X$~is called the structure sheaf of~$X$.
\endclaim

By definition a complex space $X$ is locally isomorphic to an analytic
set, so the concepts of holomorphic function on $X$, of analytic
subset, of analytic morphism, etc $\ldots$ are meaningful.  If $X$ is a
complex space, Th.~4.31 implies that $X_\sing$ is an analytic
subset of $X$. 

\claim{(5.3) Theorem and definition} For every complex space~$X$,
the set $X_\reg$ is a dense open subset of $X$, and consists of a disjoint
union of connected complex manifolds $X'_\alpha$.  Let $X_\alpha$ be the
closure of $X'_\alpha$ in $X$.  Then $(X_\alpha)$ is a locally finite
family of analytic subsets of $X$, and $X=\bigcup X_\alpha$.  The sets
$X_\alpha$ are called the global irreducible components of $X$.
\endclaim

\vbox{%
\input epsfiles/fig_2_2.tex
\vskip6mm
\centerline{{\bf Fig.~II-2} The irreducible curve $y^2=x^2(1+x)$ in $\bC^2$.}}
\vskip5mm

\noindent
Observe that the germ at a given point of a global irreducible component
can be reducible, as shows the example of the cubic curve
$\Gamma:y^2=x^2(1+x)$~; the germ $(\Gamma,0)$ has two analytic branches
$y=\pm x\,\sqrt{1+x}$, however $\Gamma\ssm\{0\}$ is easily seen to
be a connected smooth Riemann surface (the real points of $\gamma$
corresponding to $-1\le x\le 0$ form a path connecting the two branches).
This example shows that the notion of {\it global irreducible component}
is quite different from the notion of local irreducible component
introduced in~(4.4).


\proof{Proof} By definition of $X_\reg$, the connected components $X'_\alpha$
are (disjoint) complex manifolds. Let us show that the germ of $X_\alpha=
\ol X'_\alpha$ at any point $x\in X$ is analytic. We may assume that 
$(X,x)$ is a germ of analytic set $A$ in an open subset of $\bC^n$. Let 
$(A_l,x)$, $1\le l\le N$, be the irreducible components of this germ and
$U$ a neighborhood of $x$ such that $X\cap U=\bigcup A_l\cap U$. Let
$\Omega_l\subset U$ be a neighborhood of $x$ such that $A_{l,\reg}
\cap\Omega_l$ is connected and dense in $A_l\cap\Omega_l$
(Prop.~4.24). Then $A'_l:=X_\reg\cap A_l\cap\Omega_l$ equals
$(A_{l,\reg}\cap\Omega_l)\ssm\bigcup_{k\ne l}A_{l,\reg}\cap 
\Omega_l\cap A_k$. However, $A_{l,\reg}\cap\Omega_l\cap A_k$ is an 
analytic subset of $A_{l,\reg}\cap\Omega_l$, distinct from 
$A_{l,\reg}\cap\Omega_l$, otherwise $A_{l,\reg}\cap\Omega_l$
would be contained in $A_k$, thus $(A_l,x)\subset(A_k,x)$ by density. 
Remark~4.2 implies that $A'_l$ is connected and dense in $A_{l,\reg}
\cap\Omega_l$, hence in $A_l\cap\Omega_l$. Set $\Omega=\bigcap\Omega_l$ and 
let $(X_\alpha)_{\alpha\in J}$ be the family of global components
which meet $\Omega$ (i.e.\ such that $X'_\alpha\cap\Omega\ne\emptyset\,$). 
As $X_\reg\cap\Omega=\bigcup A'_l\cap\Omega$, each $X'_\alpha$, 
$\alpha\in J$, meets at least one set $A'_l$, and as $A'_l\subset X_\reg$
is connected, we have in fact $A'_l\subset X'_\alpha$. It follows that there
exists a partition $(L_\alpha)_{\alpha\in J}$ of $\{1,\ldots,N\}$ such that 
$X'_\alpha\cap\Omega=\bigcup_{l\in L_\alpha}A'_l\cap\Omega$, $\alpha\in J$.
Hence $J$ is finite, $\hbox{\rm card}\,J\le N$, and
$$X_\alpha\cap\Omega=\ol X'_\alpha\cap\Omega=\bigcup_{l\in L_\alpha}\ol A'_l
\cap\Omega=\bigcup_{l\in L_\alpha}A_l\cap\Omega$$
is analytic for all $\alpha\in J$.\qed
\endproof

\claim{(5.4) Corollary} If $A,B$ are analytic subsets in a complex space
$X$, then the closure $\ol{A\ssm B}$ is an analytic subset,
consisting of the union of all global irreducible components $A_\lambda$
of $A$ which are not contained in $B$.
\endclaim

\proof{Proof} Let $C=\bigcup A_\lambda$ be the union of these components. 
Since $(A_\lambda)$ is locally finite, $C$ is analytic.
Clearly $A\ssm B=C\ssm B=\bigcup A_\lambda\ssm B$.
The regular part $A'_\lambda$ of each $A_\lambda$ is a connected 
manifold and $A'_\lambda\cap B$ is a proper analytic subset (otherwise
$A'_\lambda\subset B$ would imply $A_\lambda\subset B)$. Thus
$A'_\lambda\ssm(A'_\lambda\cap B)$ is dense in $A'_\lambda$ which 
is dense in $A_\lambda$, so $\ol{A\ssm B}=\bigcup A_\lambda=C$.\qed
\endproof

\claim{(5.5) Theorem} For any family $(A_\lambda)$ of analytic sets in a 
complex space~$X$, the intersection $A=\bigcap A_\lambda$ is an analytic subset
of $X$. Moreover, the inter\-section is stationary on every compact
subset of $X$.
\endclaim

\proof{Proof} It is sufficient to prove the last statement, namely that every
point $x\in X$ has a neighborhood $\Omega$ such that $A\cap\Omega$ is
already obtained as a finite intersection.  However, since $\cO_{X,x}$
is Noetherian, the family of germs of finite intersections has a minimum
element $(B,x)$, $B=\bigcap A_{\lambda_j}$, $1\le j\le N$.  Let
$\tilde B$ be the union of the global irreducible components $B_\alpha$
of $B$ which contain the point $x$~; clearly $(B,x)=(\tilde B,x)$. 
For any set $A_\lambda$ in the family, the minimality of $B$ implies
$(B,x)\subset(A_\lambda,x)$.  Let $B'_\alpha$ be the regular part of
any global irreducible component $B_\alpha$ of $\tilde B$.  Then
$B'_\alpha\cap A_\lambda$ is a closed analytic subset of $B'_\alpha$
containing a non empty open subset (the intersection of $B'_\alpha$ with
some neighborhood of $x$), so we must have $B'_\alpha\cap A_\lambda=
B'_\alpha$.  Hence $B_\alpha=\ol B'_\alpha\subset A_\lambda$
for all $B_\alpha\subset\tilde B$ and all $A_\lambda$, thus 
$\tilde B\subset A=\bigcap A_\lambda$.  We infer
$$(B,x)=(\tilde B,x)\subset(A,x)\subset (B,x),$$ 
and the proof is complete.\qed
\endproof

As a consequence of these general results, it is not difficult to show that
a complex space always admits a (locally finite) stratification such that
the strata are smooth manifolds.

\claim{(5.6) Proposition} Let $X$ be a complex space. Then there is a
locally stationary increasing sequence of analytic subsets $Y_k\subset X$,
$k\in\bN$, such that $Y_0$ is a discrete set and such that $Y_k\ssm Y_{k-1}$
is a smooth $k$-dimensional complex manifold for~$k\ge 1$. Such a sequence
is called a stratification of $X$, and the sets $Y_k\ssm Y_{k-1}$ are called
the strata $($the strata may of course be empty for some indices $k<\dim X)$.
\endclaim

\proof{Proof} Let $\cF$ be the family of irreducible analytic subsets
$Z\subset X$ which can be obtained through a finite sequence of steps
of the following types:
\medskip
\item{a)} $Z$ is an irreducible component of $X\,$;
\smallskip
\item{b)} $Z$ is an irreducible component of the singular set $Z'_\sing$
of some member $Z'\in\cF\,$;
\smallskip
\item{c)} $Z$ is an irreducible component of some finite intersection of
sets $Z_j\in\cF$.
\medskip
Since $X$ has locally finite dimension and since steps b) or c) decrease
the dimension of our sets~$Z$, it is clear that $\cF$ is a locally finite
family of analytic sets in~$X$. Let $Y_k$ be the union of all sets $Z\in\cF$
of dimension $\le k$. It is easily seen that $\bigcup Y_k=X$ and
that the irreducible components of $(Y_k)_\sing$ are contained in $Y_{k-1}$
(these components are either intersections of components $Z_j\subset Y_k$ or
parts of the singular set of some component $Z\subset Y_k$, so there are in
either case obtained by step b) or c) above). Hence $Y_k\ssm Y_{k-1}$ is
a smooth manifold and it is of course $k$-dimensional, because the components
of $Y_k$ of dimension ${}<k$ are also contained in $Y_{k-1}$ by definition.
\endproof

\claim{(5.7) Theorem} Let $X$ be an irreducible complex space. Then every
non constant holomorphic function $f$ on $X$ defines an open map
$f:X\longrightarrow\bC$.
\endclaim

\proof{Proof} We show that the image $f(\Omega)$ of any neighborhood $\Omega$
of $x\in X$ contains a neighborhood of $f(x)$. Let $(X_l,x)$ be an irreducible
component of the germ $(X,x)$ (embedded in $\bC^n$) and 
$\Delta=\Delta'\times\Delta''\subset\Omega$ a polydisk 
such that the projection $\pi:X_l\cap\Delta\longrightarrow\Delta'$ is a ramified covering.
The function $f$ is non constant on the dense open manifold $X_\reg$, 
so we may select a complex line $L\subset\Delta'$ through $0$, not contained 
in the ramification locus of $\pi$, such that $f$ is non constant on the
one dimensional germ $\pi^{-1}(L)$. Therefore we can find a germ of curve
$$(\bC,0)\ni t\longmapsto\gamma(t)\in(X,x)$$
such that $f\circ\gamma$ is non constant. This implies that the image of
every neighborhood of $0\in\bC$ by $f\circ\gamma$ already contains a 
neighborhood of~$f(x)$.\qed
\endproof

\claim{(5.8) Corollary} If $X$ is a compact irreducible analytic space, then
every holomorphic function $f\in\cO(X)$ is constant.
\endclaim

In fact, if $f\in\cO(X)$ was non constant, $f(X)$ would be compact
and also open in $\bC$ by Th.~5.7, a contradiction. This result
implies immediately the following consequence.

\claim{(5.9) Theorem} Let $X$ be a complex space such that the global
holomorphic functions in $\cO(X)$ separate the points of $X$. Then
every compact analytic subset $A$ of $X$ is finite.
\endclaim

\proof{Proof} $A$ has a finite number of irreducible components $A_\lambda$
which are also compact. Every function $f\in\cO(X)$ is constant on $A_\lambda$,
so $A_\lambda$ must be reduced to a single point.\qed
\endproof

\subsection{5.2}{Coherent Sheaves over Complex Spaces}
Let $X$ be a complex space and $\cO_X$ its structure sheaf.  
Locally, $X$ can be identified to an analytic set $A$ in an open set
$\Omega\subset\bC^n$, and we have $\cO_X=\cO_\Omega/\cI_A$.
Thus $\cO_X$ is coherent over the sheaf of rings $\cO_\Omega$. It
follows immediately that $\cO_X$ is coherent over itself.
Let $\cS$ be a $\cO_X$-module.   If
$\tilde\cS$ denotes the extension of $\cS_{\restriction A}$ to
$\Omega$ obtained by setting $\tilde\cS_x=0$ for
$x\in\Omega\ssm A$, then $\tilde\cS$ is a
$\cO_\Omega$-module, and it is easily seen that $\cS_{\restriction
A}$ is coherent over $\cO_{X\restriction A}$ if and only if
$\tilde\cS$ is coherent over $\cO_\Omega$.  If $Y$
is an analytic subset of $X$, then $Y$ is locally given by an analytic
subset $B$ of $A$ and the sheaf of ideals of $Y$ in $\cO_X$ is the quotient
$\cI_Y=\cI_B/\cI_A$~; hence $\cI_Y$ is coherent.  Let us mention the
following important property of supports.

\claim{(5.10) Theorem} If $\cS$ is a coherent $\cO_X$-module, the
support of $\cS$, defined as~
$\Supp\,\cS=\{x\in X\,;\,\cS_x\ne 0\}$
is an analytic subset of $X$.
\endclaim

\proof{Proof} The result is local, thus after extending $\cS$ by $0$, we may
as well assume that $X$ is an open subset $\Omega\subset\bC^n$.  By
(3.12), there is an exact sequence of sheaves 
$$\cO_U^{\oplus p}\buildo G\over\longrightarrow\cO_U^{\oplus q}\buildo F\over\longrightarrow
\cS_{\restriction U}\longrightarrow 0$$
in a neighborhood $U$ of any point.  If $G:\cO_x^{\oplus p}\longrightarrow\cO_x^{\oplus q}$ is
surjective it is clear that the linear map $G(x):\bC^p\longrightarrow\bC^q$
must be surjective; conversely, if $G(x)$ is surjective, there is a
$q$-dimensional subspace $E\subset\bC^p$ on which the restriction of
$G(x)$ is a bijection onto $\bC^q$~; then $G_{\restriction
E}:\cO_U\otimes_\bC E\longrightarrow\cO_U^{\oplus q}$ is bijective near $x$ and $G$ is
surjective.  The support of $\cS_{\restriction U}$ is thus equal to the
set of points $x\in U$ such that all minors of $G(x)$ of order $q$
vanish.\qed
\endproof

\section{6}{Analytic Cycles and Meromorphic Functions}
\subsection{6.1}{Complete Intersections}
Our goal is to study in more details the dimension of a subspace given 
by a set of equations. The following proposition is our starting point.

\claim{(6.1) Proposition} Let $X$ be a complex space of pure dimension
$p$ and $A$ an analytic subset of $X$ with $\codim_X A\ge 2$. Then every
function $f\in\cO(X\ssm A)$ is locally bounded near $A$.
\endclaim

\proof{Proof} The statement is local on $X$, so we may assume that $X$ is 
an irreducible germ of analytic set in $(\bC^n,0)$. Let $(A_k,0)$ be the 
irreducible components of $(A,0)$. By a reasoning analogous to that of
Prop.~4.26, we can choose coordinates $(z_1,\ldots,z_n)$ on $\bC^n$
such that all projections
$$\eqalign{
\pi:{}&z\longmapsto(z_1,\ldots,z_p),~~~p=\dim X,\cr
\pi_k:{}&z\longmapsto(z_1,\ldots,z_{p_k}),~~p_k=\dim A_k\,,\cr}$$
define ramified coverings $\pi:X\cap\Delta\longrightarrow\Delta'$,
$\pi_k:A_k\cap\Delta\longrightarrow\Delta'_k$. By construction $\pi(A_k)\subset\Delta'$
is contained in the set $B_k$ defined by some Weierstrass polynomials in the
variables $z_{p_k+1},\ldots,z_p$ and $\codim_{\Delta'}B_k=p-p_k\ge 2$. Let $S$
be the ramification locus of $\pi$ and $B=\bigcup B_k$. We have
$\pi(A\cap\Delta)\subset B$. For $z'\in\Delta'\ssm(S\cup B)$,
we let
$$\sigma_k(z')=\hbox{\rm elementary symmetric function of degree~}k~
\hbox{\rm in~}f(z',z''_\alpha),$$
where $(z',z''_\alpha)$ are the $q$ points of $X$ projecting on $z'$. Then
$\sigma_k$ is holomorphic on $\Delta'\ssm(S\cup B)$ and locally 
bounded near every point of $S\ssm B$, thus $\sigma_k$ extends
holomorphically to $\Delta'\ssm B$ by Remark~4.2. Since $\codim B\ge 2$,
$\sigma_k$ extends to $\Delta'$ by Cor. 1.4.5. Now, $f$ satisfies 
$f^q-\sigma_1f^{q-1}+\ldots+(-1)^q\sigma_q=0$, thus $f$ is locally
bounded on $X\cap\Delta$.\qed
\endproof

\claim{(6.2) Theorem} Let $X$ be an irreducible complex space and $f\in\cO(X)$,
$f\not\equiv 0$. Then $f^{-1}(0)$ is empty or of pure dimension $\dim X-1$.
\endclaim

\proof{Proof} Let $A=f^{-1}(0)$. By Prop.~4.26, we know that
$\dim A\le\dim X-1$. If $A$ had an irreducible branch $A_j$ of dimension
$\le\dim X-2$, then in virtue of Prop.~6.1 the function $1/f$
would be bounded in a neighborhood of $A_j\ssm\bigcup_{k\ne j}A_k$,
a contradiction.\qed
\endproof

\claim{(6.3) Corollary} If $f_1,\ldots,f_p$ are holomorphic functions on an
irreducible complex space $X$, then all irreducible components of
$f_1^{-1}(0)\cap\ldots\cap f_p^{-1}(0)$ have codimension~$\ge p$.\qed
\endclaim

\claim{(6.4) Definition} Let $X$ be a complex space of pure dimension
$n$ and $A$ an analytic subset of $X$ of pure dimension. Then $A$ is
said to be a local $($set theoretic$)$ complete intersection in $X$ if
every point of $A$ has a neighborhood $\Omega$ such that
$$A\cap\Omega=\{x\in\Omega\,;\,f_1(x)=\ldots=f_p(x)=0\}$$
with exactly $p=\codim\,A$ functions $f_j\in\cO(\Omega)$.
\endclaim

\claim{(6.5) Remark} \rm As a converse to Th.~6.2, one may ask
whether every hypersurface $A$ in $X$ is locally defined by a single
equation $f=0$. In general the answer is negative. A simple
counterexample for $\dim X=3$ is obtained with the singular quadric
$X=\{z_1z_2+z_3z_4=0\}\subset\bC^4$ and the plane
$A=\{z_1=z_3=0\}\subset X$. Then $A$ cannot be defined by 
a single equation $f=0$ near the origin, otherwise the plane
$B=\{z_2=z_4=0\}$ would be such that
$$f^{-1}(0)\cap B=A\cap B=\{0\},$$
in contradiction with Th.~6.2 (also, by Exercise~10.11, we would
get the inequality $\codim_X A\cap B\le 2$). However, the answer is
positive when $X$ is a manifold:
\endclaim

\claim{(6.6) Theorem} Let $M$ be a complex manifold with $\dim_\bC M=n$, let
$(A,x)$ be an analytic germ of pure dimension $n-1$ and let $A_j$,
$1\le j\le N$, be its irreducible components.
\smallskip
\item{\rm a)} The ideal of $(A,x)$ is a principal ideal
$\cI_{A,x}=(g)$ where $g$ is a product of irreducible germs $g_j$ 
such that $\cI_{A_j,x}=(g_j)$.
\smallskip
\item{\rm b)} For every $f\in\cO_{M,x}$ such that $f^{-1}(0)\subset(A,x)$,
there is a unique decompo\-sition $f=ug_1^{m_1}\ldots g_N^{m_N}$
where $u$ is an invertible germ and $m_j$ is the order of vanishing
of $f$ at any point $z\in A_{j,\reg}\ssm\bigcup_{k\ne j}A_k$.\smallskip
\endclaim

\proof{Proof} a) In a suitable local coordinate system centered at $x$, the 
projection $\pi:\bC^n\longrightarrow\bC^{n-1}$ realizes all $A_j$ as ramified
coverings
$$\pi:A_j\cap\Delta\longrightarrow\Delta'\subset\bC^{n-1},~~~~
\hbox{ramification locus}=S_j\subset\Delta'.$$
The function
$$g_j(z',z_n)=\prod_{w\in A_j\cap\pi^{-1}(z')}(z_n-w_n),~~~~
z'\in\Delta'\ssm S_j$$
extends into a holomorphic function in $\cO_{\Delta'}[z_n]$ and is
irreducible at $x$. Set
$g=\prod g_j\in\cI_{A,x}$. For any $f\in\cI_{A,x}$, the Weierstrass
division theorem yields $f=gQ+R$ with $R\in\cO_{n-1}[z_n]$
and deg$\,R<\deg\,g$. As $R(z',z_n)$ vanishes when $z_n$ is equal
to $w_n$ for each point $w\in A\cap\pi^{-1}(z')$, $R$ has
exactly $\deg\,g$ roots when $z'\in\Delta'\ssm\big(
\bigcup S_j\cup\bigcup\pi(A_j\cap A_k)\big)$, so $R=0$. 
Hence $\cI_{A,x}=(g)$ and similarly $\cI_{A_j,x}=(g_j)$.
Since $\cI_{A_j}$ is coherent, $g_j$ is also a generator of
$\cI_{A_j,z}$ for $z$ near $x$ and we infer that $g_j$ has order $1$
at any regular point $z\in A_{j,\reg}$.
\medskip
\noindent{\rm b)} As $\cO_{M,x}$ is factorial, any $f\in\cO_{M,x}$ can be written 
$f=u\,g_1^{m_1}\ldots g_N^{m_N}$ where $u$ is either invertible or a product
of irreducible elements distinct from the $g_j$'s. In the latter case
the hypersurface $u^{-1}(0)$ cannot be contained in $(A,x)$,
otherwise it would be a union of some of the components $A_j$ and
$u$ would be divisible by some $g_j$. This proves b).\qed
\endproof

\claim{(6.7) Definition} Let $X$ be an complex space of pure
dimension $n$. 
\smallskip
\item{\rm a)} An analytic $q$-cycle $Z$ on $X$ is a formal linear 
combination $\sum\lambda_j A_j$ where $(A_j)$ is a locally finite family of
irreducible analytic sets of dimension $q$ in $X$ and $\lambda_j\in\bZ$.
The support of $Z$ is $|Z|=\bigcup_{\lambda_j\ne 0}A_j$.
The group of all $q$-cycles on $X$ is denoted $\Cycl^q(X)$.
Effective $q$-cycles are elements of the subset $\Cycl^q_+(X)$ of
cycles such that all coefficients $\lambda_j$ are $\ge 0~;$ rational,
real cycles are cycles with coefficients $\lambda_j\in\bQ,~\bR$.
\smallskip
\item{\rm b)} An analytic $(n-1)$-cycle is called a $($Weil$\,)$ divisor,
and we set 
$$\Div(X)=\Cycl^{n-1}(X).$$
\item{\rm c)} Assume that $\dim X_\sing\le n-2$. If $f\in\cO(X)$
does not vanish identically on any irreducible component of $X$,
we associate to $f$ a divisor 
$$\div(f)=\sum m_j A_j\in\Div_+(X)$$
in the following way: the components $A_j$ are the irreducible components
of $f^{-1}(0)$ and the coefficient $m_j$ is the vanishing order of $f$ at
every regular point in $X_\reg\cap A_{j,\reg}\ssm\bigcup_{k\ne j}A_k$.
It is clear that we have 
$$\div(fg)=\div(f)+\div(g).$$
\item{\rm d)} A Cartier divisor is a divisor $D=\sum\lambda_jA_j$ 
that is equal locally to a $\bZ$-linear combination of divisors of the 
form $\div(f)$.\smallskip
\endclaim

It is easy to check that the collection of abelian groups $\Cycl^q(U)$ 
over all open sets $U\subset X$, together with the
obvious restriction morphisms, satisfies axioms (1.4) of sheaves;
observe however that the restriction of an irreducible component $A_j$
to a smaller open set may subdivide in several components.  Hence we
obtain sheaves of abelian groups $\Cycl^q$ and $\Div=\Cycl^{n-1}$ 
on $X$.  The stalk $\Cycl^q_x$ is the free
abelian group generated by the set of irreducible germs of
$q$-dimensional analytic sets at the point $x$.  These sheaves carry a
natural partial ordering determined by the subsheaf of positive elements
$\Cycl^q_+$.  We define the sup and inf of two analytic cycles 
$Z=\sum\lambda_j A_j$, $Z'=\sum\mu_j A_j$ by
$$\sup\{Z,Z'\}=\sum\sup\{\lambda_j,\mu_j\}\,A_j,~~
\inf\{Z,Z'\}=\sum\inf\{\lambda_j,\mu_j\}\,A_j\,;\leqno(6.8)$$
it is clear that these operations are compatible with restrictions, i.e.\ they
are defined as sheaf operations.

\claim{(6.9) Remark} \rm When $X$ is a manifold, Th.~6.6 shows that
every effective $\bZ$-divisor is locally the divisor of a holomorphic
function; thus, for manifolds, the concepts of Weil and Cartier
divisors coincide.  This is not always the case in general: in Example
6.5, one can show that $A$ is not a Cartier divisor (exercise 10.?).
\endclaim

\subsection{6.2}{Divisors and Meromorphic Functions}
Let $X$ be a complex space. For $x\in X$, let $\cM_{X,x}$
be the ring of quotients of $\cO_{X,x}$, i.e.\ the set of formal quotients
$g/h$, $g,h\in\cO_{X,x}$, where $h$ is not a zero divisor in $\cO_{X,x}$,
with the identification $g/h=g'/h'$ if $gh'=g'h$. We consider the
disjoint union
$$\cM_X=\coprod_{x\in X}\cM_{X,x}\leqno(6.10)$$
with the topology in which the open sets open sets are unions of sets of 
the type $\{G_x/H_x\,;\,x\in V\}\subset\cM_X$ where $V$ is open in $X$
and $G,H\in\cO_X(V)$. Then $\cM_X$ is a sheaf over $X$, and the sections 
of $\cM_X$ over an open set $U$ are called {\it meromorphic functions} on $U$.
By definition, these sections can be represented locally as quotients of 
holomorphic functions, but there need not exist such a global representation
on~$U$.

A point $x\in X$ is called a {\it pole} of a meromorphic function $f$ on
$X$ if $f_x\notin\cO_{X,x}$.  Clearly, the set $P_f$ of poles of $f$ is
a closed subset of $X$ with empty interior: if $f=g/h$ on $U$, then
$h\not\equiv 0$ on any irreducible component and $P_f\cap U\subset
h^{-1}(0)$.  For $x\notin P_f$, one can speak of {\it the value}
$f(x)$.  If the restriction of $f$ to $X_\reg\ssm P_f$ does
not vanish identically on any irreducible component of $(X,x)$, then
$1/f$ is a meromorphic function in a neighborhood of $x$~; the set of
poles of $1/f$ will be denoted $Z_f$ and called the {\it zero set} 
of $f$.  If $f$ vanishes on some connected open subset of 
$X_\reg\ssm P_f$, then $f$ vanishes identically (outside
$P_f$) on the global irreducible component $X_\alpha$ containing this
set; we agree that these components $X_\alpha$ are contained in $Z_f$. 
For every point $x$ in the complement of $Z_f\cap P_f$, we have either
$f_x\in\cO_{X,x}$ or $(1/f)_x\in\cO_{X,x}$, thus $f$ defines a
holomorphic map $X\ssm(Z_f\cap P_f)\longrightarrow\bC\cup\{\infty\}=\bP^1$ with
values in the projective line.  In general, no value (finite or
infinite) can be assigned to $f$ at a point $x\in Z_f\cap P_f$, as
shows the example of the function $f(z)=z_2/z_1$ in $\bC^2$.  The set
$Z_f\cap P_f$ is called the {\it indeterminacy set} of $f$. 

\claim{(6.11) Theorem} For every meromorphic function $f$ on $X$,
the sets $P_f$, $Z_f$ and the indeterminacy set $Z_f\cap P_f$
are analytic subsets.
\endclaim

\proof{Proof} Let $\cJ_x$ be the ideal of germs $u\in\cO_{X,x}$ such that 
$uf_x\in\cO_{X,x}$. Let us write $f=g/h$ on a small open set $U$. 
Then $\cJ_{\restriction U}$ appears as the projection on the first 
factor of the sheaf of relations $\cR(g,h)\subset\cO_U\times\cO_U$, 
so $\cJ$ is a coherent sheaf of ideals. Now
$$P_f=\big\{x\in X\,;\,\cJ_x=\cO_{X,x}\big\}=\Supp\,\cO_X/\cJ,$$
thus $P_f$ is analytic by Th.~5.10. Similarly, the projection
of $\cR(g,h)$ on the second factor defines a sheaf of ideals $\cJ'$
such that $Z_f=\Supp\,\cO_X/\cJ'$.\qed
\endproof

When $X$ has pure dimension $n$ and $\dim X_\sing\le n-2$,
Def.~6.7~c) can be extended to meromorphic functions:
if $f=g/h$ locally, we set
$$\div(f)=\div(g)-\div(h).\leqno(6.12)$$
By 6.7~c), we immediately see that this definition does not depend
on the choice of the local representant $g/h$. Furthermore,
{\it Cartier divisors} are precisely those divisors which are 
associated locally to meromorphic functions.\medskip

Assume from now on that $M$ is a connected $n$-dimensional complex
manifold.  Then, for every point $x\in M$, the ring
$\cO_{M,x}\simeq\cO_n$ is factorial.  This property makes the study of
meromorphic functions much easier. 

\claim{(6.13) Theorem} Let $f$ be a non zero meromorphic function on a
manifold $M$, with $\dim_\bC M=n$.  Then the sets $Z_f$, $P_f$ are purely
$(n-1)$-dimensional, and the indeterminacy set $Z_f\cap P_f$ is purely
$(n-2)$-dimensional.
\endclaim

\proof{Proof} For every point $a\in M$, the germ $f_a$ can be written $g_a/h_a$ 
where $g_a,h_a\in\cO_{M,a}$ are relatively prime holomorphic germs. By
Th.~1.12, the germs $g_x$, $h_x$ are still relatively prime for $x$ in a 
neighborhood $U$ of $a$. Thus the ideal $\cJ$ associated to $f$ coincides 
with $(h)$ on $U$, and we have
$$P_f\cap U=\Supp\,\cO_U/(h)=h^{-1}(0),~~~~Z_f\cap U=g^{-1}(0).$$
Th.~6.2 implies our contentions: if $g_\lambda$ and $h_\mu$ are the
irreducible components of $g,h$, then $Z_f\cap P_f=\bigcup g_\lambda^{-1}(0)
\cap h_\mu^{-1}(0)$ is $(n-2)$-dimensional. As we will see in the next section,
Th.~6.13 does not hold on an arbitrary complex space.\qed
\endproof

Let $(A_j)$, resp.\ $(B_j)$, be the global irreducible components of $Z_f$,
resp.\ $P_f$. In a neighborhood $V_j$ of the $(n-1)$-dimensional
analytic set
$$A'_j=A_j\ssm\big(P_f\cup\bigcup_{k\ne j}A_k)$$
$f$ is holomorphic and $V\cap f^{-1}(0)=A'_j$. As $A'_{j,\reg}$ is
connected, we must have $\div(f_{\restriction V_j})=m_jA'_j$ for some
constant multiplicity $m_j$ equal to the vanishing order of $f$
along $A'_{j,\reg}$. Similarly, $1/f$ is holomorphic in
a neighborhood $W_j$ of
$$B'_j=B_j\ssm\big(Z_f\cup\bigcup_{k\ne j}B_k)$$
and we have $\div(f_{\restriction V})=-p_jB'_j$ where $p_j$ is
the vanishing order of $1/f$ along $B'_{j,\reg}$. At a point $x\in M$
the germs $A_{j,x}$ and $B_{j,x}$ may subdivide in irreducible
local components $A_{j,\lambda,x}$ and $B_{j,\lambda,x}$. If
$g_{j,\lambda}$ and $h_{j,\lambda}$ are local generators of the
corresponding ideals, we may a priori write
$$f_x=u\,g/h~~~\hbox{\rm where}~~~g=\prod g_{j,\lambda}^{m_{j,\lambda}},~~~
h=\prod h_{j,\lambda}^{p_{j,\lambda}}$$
and where $u$ is invertible. Then necessarily $m_{j,\lambda}=m_j$ and
$p_{j,\lambda}=p_j$ for all $\lambda$, and we see that the global
divisor of $f$ on $M$ is
$$\div(f)=\sum m_j A_j-\sum p_j B_j.\leqno(6.14)$$
Let us denote by $\cM^\star$ the multiplicative sheaf of germs of
non zero meromorphic functions, and by $\cO^\star$ the sheaf of germs
of invertible holomorphic functions. Then we have an exact sequence
of sheaves
$$1\longrightarrow\cO^\star\longrightarrow\cM^\star\buildo\div\over\longrightarrow\Div\longrightarrow 0.
\leqno(6.15)$$
Indeed, the surjectivity of div is a consequence of Th.~6.6.
Moreover, any meromorphic function that has a positive divisor must be
holomorphic by the fact that $\cO_n$ is factorial. Hence a meromorphic
function $f$ with $\div(f)=0$ is an invertible holomorphic function.

\section{7}{Normal Spaces and Normalization}
\subsection{7.1}{Weakly Holomorphic Functions}
The goal of this section is to show that the singularities
of $X$ can be studied by enlarging the structure sheaf
$\cO_X$ into a sheaf $\tilde\cO_X$ of so-called
weakly holomorphic functions. 

\claim{(7.1) Definition} Let $X$ be a complex space. A weakly
holomorphic function $f$ on $X$ is a holomorphic function on $X_\reg$
such that every point of $X_\sing$ has a neighborhood $V$ for
which $f$ is bounded on $X_\reg\cap V$.  We denote by
$\tilde\cO_{X,x}$ the ring of germs of weakly holomorphic functions
over neighborhoods of $x$ and $\tilde\cO_X$ the associated sheaf.
\endclaim

Clearly, $\tilde\cO_{X,x}$ is a ring containing $\cO_{X,x}$.
If $(X_j,x)$ are the irreducible components of $(X,x)$, 
there is a fundamental system of neighborhoods $V$ of $x$ such that
$X_\reg\cap V$ is a disjoint union of connected open sets
$$X_{j,\reg}\cap V\ssm\bigcup_{k\ne j}X_k\cap X_{j,\reg}\cap V$$
which are dense in $X_{j,\reg}\cap V$. Therefore any bounded
holomorphic function on $X_\reg\cap V$ extends to
each component $X_{j,\reg}\cap V$ and we see that
$$\tilde\cO_{X,x}=\bigoplus\tilde\cO_{X_j,x}.$$
The first important fact is that weakly holomorphic functions are always
meromorphic and possess ``universal denominators".

\claim{(7.2) Theorem} For every point $x\in X$, there is a neighborhood
$V$ of $x$ and $h\in\cO_X(V)$ such 
that $h^{-1}(0)$ is nowhere dense in $V$ and $h_y\tilde\cO_{X,y}\subset
\cO_{X,y}$ for all $y\in V$~; such a function $h$ is called a universal
denominator on $V$. In particular $\smash{\tilde\cO}_X$ is contained in
the ring $\cM_X$ of meromorphic functions.
\endclaim

\proof{Proof} First assume that $(X,x)$ is irreducible and that we have 
a ramified covering $\pi:X\cap\Delta\longrightarrow\Delta'$ with ramification
locus $S$. We claim that the discriminant $\delta(z')$ of a primitive
element $u(z'')=c_{d+1}z_{d+1}+\cdots+c_nz_n$ is a universal denominator
on $X\cap\Delta$. To see this, we imitate the proof of Lemma~4.15.
Let $f\in\tilde\cO_{X,y}$, $y\in X\cap\Delta$. Then we solve the 
equation
$$f(z)=\sum_{0\le j\le q}b_j(z')u(z'')^j$$
in a neighborhood of $y$. For $z'\in\Delta'\ssm S$, 
let us denote by $(z',z''_\alpha)$, $1\le\alpha\le q$, the points
in the fiber $X\cap\pi^{-1}(z')$. Among these, only $q'$ are close to
$y$, where $q'$ is the sum of the sheet
numbers of the irreducible components of $(X,y)$ by the projection $\pi$.
The other points $(z',z''_\alpha)$, say $q'<\alpha\le q$, are in 
neighborhoods of the points of $\pi^{-1}(y')\ssm\{y\}$.
We take $\big(b_j(z')\big)$ to be the solution of the linear system
$$\sum_{0\le j\le q}b_j(z')u(z''_\alpha)^j=
\cases{f(z',z''_\alpha)&for~~$1\le\alpha\le q'$,\cr
                      0&for~~$q'<\alpha\le n$.\cr}$$
The solutions $b_j(z')$ are holomorphic on $\Delta'\ssm S$ near $y'$.
Since the determinant is $\delta(z')^{1/2}$, we see that $\delta b_j$
is bounded, thus $\delta b_j\in\cO_{\Delta',y'}$ and $\delta_y f\in\cO_{X,y}$.

Now, assume that $(X,x)\subset(\bC^n,0)$ has irreducible components $(X_j,x)$.
We can find for each $j$ a neighborhood $\Omega_j$ of $0$ in $\bC^n$ and
a function $\delta_j\in\cO_n(\Omega_j)$ which is a universal
denominator on $X_j\cap\Omega_j$. After adding to $\delta_j$ 
a function which is identically zero on $(X_j,x)$ and non zero on $(X_k,x)$,
$k\ne j$, we may assume that $\delta_j^{-1}(0)\cap X_k\cap\Omega$
is nowhere dense in $X_k\cap\Omega$ for all $j$ and $k$ and some
small $\Omega\subset\bigcap\Omega_j$. Then $\delta=\prod\delta_j$
is a universal denominator on each component $X_j\cap\Omega$.
For some possibly smaller $\Omega$, select a function
$v_j\in\cO_n(\Omega)$ such that $v_j$ vanishes identically on 
$\bigcup_{k\ne j}X_k\cap\Omega$ and $v_j^{-1}(0)$ is nowhere dense in 
$X_j\cap\Omega$, and set $h=\delta\sum v_k$. For any germ 
$f\in\cO_{X,y}$, $y\in X\cap\Omega$, there is a germ 
$g_j\in\cO_{\Omega,y}$ with $\delta f=g_j$ on $(X_j,y)$. We have
$h=\delta v_j$ on $X_j\cap\Omega$, so $h^{-1}(0)$ is nowhere
dense in $X\cap\Omega$ and
$$hf=v_j\delta f=v_j g_j=\sum v_k g_k~~~\hbox{\rm on each}~~(X_j,y).$$
Since $\sum v_k g_k\in\cO_{\Omega,y}$, we get
$h\tilde\cO_{X,y}\subset\cO_{X,y}$.\qed
\endproof

\claim{(7.3) Theorem} If $(X,x)$ is irreducible, $\tilde\cO_{X,x}$ is the
integral closure of $\cO_{X,x}$ in its quotient field $\cM_{X,x}$.
Moreover, every germ $f\in\tilde\cO_{X,x}$ admits a limit
$$\lim_{X_\reg\ni z\to x}f(z).$$
\endclaim

Observe that $\cO_{X,x}$ is an entire ring, so the ring of quotients 
$\cM_{X,x}$ is actually a field. A simple illustration of the theorem
is obtained with the irreducible germ of curve $X:z_1^3=z_2^2$
in $(\bC^2,0)$. Then $X$ can be parametrized by $z_1=t^2$, $z_2=t^3$,
$t\in\bC$, and $\cO_{X,0}=\bC\{z_1,z_2\}/(z_1^3-z_2^2)=\bC\{t^2,t^3\}$
consists of all convergent series $\sum a_nt^n$ with $a_1=0$.
The function $z_2/z_1=t$ is weakly holomorphic on $X$ and
satisfies the integral equation $t^2-z_1=0$. Here we have 
$\tilde\cO_{X,0}=\bC\{t\}$.

\proof{Proof} a) Let $f=g/h$ be an element in $\cM_{X,x}$ satisfying an 
integral equation
$$f^m+a_1f^{m-1}+\ldots+a_m=0,~~~~a_k\in\cO_{X,x}.$$
Set $A=h^{-1}(0)$. Then $f$ is holomorphic on $X\ssm A$ near $x$, 
and Lemma~4.10 shows that $f$ is bounded on a neighborhood of $x$.
By Remark~4.2, $f$ can be extended as a holomorphic function on 
$X_\reg$ in a neighborhood of $x$, thus $f\in\tilde\cO_{X,x}$.
\medskip
\noindent{\rm b)} Let $f\in\tilde\cO_{X,x}$ and let $\pi:X\cap\Delta\longrightarrow
\Delta'$ be a ramified covering in a neighborhood of $x$, with ramification
locus~$S$. As in the proof of Th.~6.1, $f$ satisfies an equation
$$f^q-\sigma_1f^{q-1}+\cdots+(-1)^q\sigma_q=0,~~~~\sigma_k\in\cO(\Delta')~;$$
indeed the elementary symmetric functions $\sigma_k(z')$ are holomorphic
on $\Delta'\ssm S$ and boun\-ded, so they extend holomorphically 
to $\Delta'$. Hence $\tilde\cO_{X,x}$ is integral 
over $\cO_{X,x}$ and we already know that $\tilde\cO_{X,x}\subset
\cM_{X,x}$.\medskip

\noindent{\rm c)} Finally, the cluster set $\bigcap_{V\ni x}
\ol{f(X_\reg\cap V)}$ is connected, because there is a fundamental
system of neighborhoods $V$ of $x$ such that $X_\reg\cap V$ is
connected, and any intersection of a decreasing sequence of compact
connected sets is connected.  However the limit set is contained in the
finite set of roots of equation b) at point $x'\in\Delta'$, so it must
be reduced to one element.\qed
\endproof

\subsection{7.2}{Normal Spaces}
Normal spaces are spaces for which all weakly holomorphic functions
are actually holomorphic. These spaces will be seen later to have
``simpler" singularities than general analytic spaces.

\claim{(7.4) Definition} A complex space $X$ is said to be normal
at a point $x$ if $(X,x)$ is irreducible and $\tilde\cO_{X,x}=\cO_{X,x}$,
that is, $\cO_{X,x}$ is integrally closed in its field of quotients.
The set of normal $($resp.\ non-normal$)$ points will be denoted 
$X_\norm$ $($resp.\ $X_\nn)$. The space $X$ itself
is said to be normal if $X$ is normal at every point.
\endclaim

Observe that any regular point $x$ is normal: in fact $\cO_{X,x}
\simeq\cO_n$ is then factorial, hence integrally closed. Therefore
$X_\nn\subset X_\sing$.

\claim{(7.5) Theorem} The non-normal set $X_\nn$ is
an analytic subset of $X$. In particular, $X_\norm$ is open
in $X$.
\endclaim

\proof{Proof} We give here a beautifully simple proof due to \cite{Grauert
and Remmert 1984}. Let $h$ be a universal denominator on a neighborhood
$V$ of a given point and let $\cI=\sqrt{h\cO_X}$ be the sheaf of ideals of 
$h^{-1}(0)$ by Hilbert's Nullstellensatz. Finally, let 
$\cF=\hom_\cO(\cI,\cI)$ be the
sheaf of $\cO_X$-endomorphisms of $\cI$. Since $\cI$ is coherent,
so is $\cF$ (cf.\ Exercise~10.?). Clearly, the homotheties of $\cI$
give an injection $\cO_X\subset\cF$ over $V$. 
We claim that there is a natural injection
$\cF\subset\tilde\cO_X$. In fact, any endomorphism of $\cI$ yields
by restriction a homomorphism $h\cO_X\longrightarrow\cO_X$, and by $\cO_X$-linearity
such a homomorphism is obtained by multiplication by an element in 
$h^{-1}\cO_X$. Thus $\cF\subset h^{-1}\cO_X\subset\cM_X$. Since
each stalk $\cI_x$ is a finite $\cO_{X,x}$-module containing non-zero
divisors, it follows that that any meromorphic germ $f$ such that
$f\cI_x\subset\cI_x$ is integral over $\cO_{X,x}$ (\cite{Lang~1965},
Chapter~IX, \S~1), hence $\cF_x\subset\tilde\cO_{X,x}$. Thus we have
inclusions $\cO_X\subset\cF\subset\tilde\cO_X$. Now, we assert that
$$X_\nn\cap V=\{x\in V\,;\,\cF_x\ne\cO_{X,x}\}=\cF/\cO_X.$$
This will imply the theorem by 5.10. To prove the equality, we first
observe that $\cF_x\ne\cO_{X,x}$ implies $\tilde\cO_{X,x}\ne\cO_{X,x}$,
thus $x\in X_\nn$. Conversely, assume that $x$ is non
normal, that is, $\tilde\cO_{X,x}\ne\cO_{X,x}$. Let $k$ be the
smallest integer such that $\cI^k_x\tilde\cO_{X,x}\subset\cO_{X,x}$~;
such an integer exists since $\cI^l_x\tilde\cO_{X,x}\subset 
h\tilde\cO_{X,x}\subset\cO_{X,x}$ for $l$ large. Then there is 
an element $w\in\cI^{k-1}_x\tilde\cO_{X,x}$
such that $w\notin\cO_{X,x}$. We have $w\cI_x\subset\cO_{X,x}$~; moreover,
as $w$ is locally bounded near $X_\sing$, any germ $wg$ in $w\cI_x$
satisfies $\lim w(z)g(z)=0$ when $z\in X_\reg$ tends to a point of 
the zero variety $h^{-1}(0)$ of $\cI_x$. Hence $w\cI_x\subset\cI_x$, 
i.e.\ $w\in\cF_x$, but $w\notin\cO_{X,x}$, so $\cF_x\ne\cO_{X,x}$.\qed
\endproof

\claim{(7.6) Theorem} If $x\in X$ is a normal point, then $(X_\sing,x)$
has codimension at least 2 in $(X,x)$.
\endclaim

\proof{Proof} We suppose that $\Sigma=X_\sing$ has codimension 1 in a
neighborhood of $x$ and try to get a contradiction. By restriction to a 
smaller neighborhood, we may assume that $X$ itself
is normal and irreducible (since $X_\norm$ is open), $\dim X=n$, that
$\Sigma$ has pure dimension $n-1$ and that the ideal sheaf $\cI_\Sigma$ has
global generators $(g_1,\ldots,g_k)$. Then $\Sigma\subset\bigcup g^{-1}_j(0)$~;
both sets have pure dimension $n-1$ and thus singular sets of
dimension $\le n-2$. Hence there is a point $a\in \Sigma$ that is
regular on $\Sigma$ and on $\bigcup g^{-1}_j(0)$, in particular there is
a neighborhood $V$ of $a$ such that $g^{-1}_1(0)\cap V=\ldots=
g^{-1}_k(0)\cap V=\Sigma\cap V$ is a smooth $(n-1)$-dimensional
manifold. Since $\codim_X\Sigma=1$ and $a$ is a singular point of $X$,
$\cI_{\Sigma,a}$ cannot have less than 2 generators in $\cO_{X,a}$ by 
Cor.~4.33. Take $(g_1,\ldots,g_l)$, $l\ge 2$, to be a minimal subset 
of generators. Then $f=g_2/g_1$ cannot belong to
$\cO_{X,a}$, but $f$ is holomorphic on $V\ssm\Sigma$. We may
assume that there is a sequence $a_\nu\in V\ssm\Sigma$ converging to $a$
such that $f(a_\nu)$ remains bounded (otherwise reverse $g_1$ and $g_2$
and pass to a subsequence). Since $g^{-1}_1(0)\cap V=\Sigma\cap V$, 
Hilbert's Nullstellensatz gives an integer $m$ such that 
$\cI_{\Sigma,a}^m\subset g_1\cO_{X,a}$, hence
$f_a\cI_{\Sigma,a}^m\subset\cO_{X,a}$. We take $m$ to be the smallest
integer such that the latter inclusion holds. Then there is a product
$g^\alpha=g_1^{\alpha_1}\ldots g_l^{\alpha_l}$ with $|\alpha|=m-1$
such that $fg^\alpha\notin\cO_{X,a}$ but $fg^\alpha g_j\in\cO_{X,a}$
for each $j$. Since the sequence $f(a_\nu)$ is bounded we conclude
that $fg^\alpha g_j$ vanishes at $a$. The zero set of this function
has dimension $n-1$ and is contained in $\bigcup g^{-1}_k(0)\cap V
=\Sigma\cap V$ so it must contain the germ $(\Sigma,a)$. Hence
$fg^\alpha g_j\in\cI_{\Sigma,a}$ and $fg^\alpha\cI_{\Sigma,a}\subset
\cI_{\Sigma,a}$. As $\cI_{\Sigma,a}$ is a finitely generated
$\cO_{X,a}$-module, this implies $fg^\alpha\in\tilde\cO_{X,a}=
\cO_{X,a}$, a contradiction.\qed
\endproof

\claim{(7.7) Corollary} A complex curve is normal if and only if it is regular.
\endclaim

\claim{(7.8) Corollary} Let $X$ be a normal complex space and
$Y$ an analytic subset of $X$ such that $\dim(Y,x)\le\dim(X,x)-2$
for any $x\in X$. Then any holo\-mor\-phic function on $X\ssm Y$
can be extended to a holomorphic function on~$X$.
\endclaim

\proof{Proof} By Cor. 1.4.5, every holomorphic function $f$ on $X_\reg
\ssm Y$ extends to~$X_\reg$.  Since $\codim X_\sing
\ge 2$, Th.~6.1 shows that $f$ is locally bounded near~$X_\sing$.
Therefore $f$ extends to $X$ by definition of a normal space.\qed
\endproof

\subsection{7.3}{The Oka Normalization Theorem}
The important normalization theorem of \cite{Oka 1950} shows that 
$\tilde\cO_X$ can be used to define the structure sheaf of a new 
analytic space $\tilde X$ which is normal and is obtained by
``simplifying" the singular set of $X$. More precisely:

\claim{(7.9) Definition} Let $X$ be a complex space. A normalization
$(Y,\pi)$ of $X$ is a normal complex space $Y$ together with a holomorphic 
map $\pi:Y\longrightarrow X$ such that the following conditions are satisfied.
\smallskip
\item{\rm a)} $\pi:Y\longrightarrow X$ is proper and has finite fibers$;$
\smallskip
\item{\rm b)} if $\Sigma$ is the set of singular points of $X$ and
$A=\pi^{-1}(\Sigma)$, then $Y\ssm A$ is dense in $Y$ and
$\pi:Y\ssm A\longrightarrow X\ssm\Sigma=X_\reg$ is an analytic
isomorphism.\smallskip
\endclaim

It follows from b) that $Y\ssm A\subset Y_\reg$. Thus
$Y$ is obtained from $X$ by a suitable ``modification" of its singular
points. Observe that $Y_\reg$ may be larger than $Y\ssm A$,
as is the case in the following two examples.

\claim{(7.10) Examples} {\smallskip\rm
\noindent{a)} Let $X=\bC\times\{0\}\cup\{0\}\times\bC$ be the complex
curve $z_1z_2=0$ in $\bC^2$. Then the normalization of $X$ is the disjoint
union $Y=\bC\times\{1,2\}$ of two copies of~$\bC$, 
with the map $\pi(t_1)=(t_1,0)$, $\pi(t_2)=(0,t_2)$. The set 
$A=\pi^{-1}(0,0)$ consists of exactly two points.
\smallskip
\noindent{b)} The cubic curve $X:z_1^3=z_2^2$ is normalized by the map
$\pi:\bC\longrightarrow X$, $t\longmapsto(t^2,t^3)$. Here $\pi$ is a
homeomorphism but $\pi^{-1}$ is not analytic at~$(0,0)$.\qed}
\endclaim

We first show that the normalization is essentially unique up to 
isomorphism and postpone the proof of its existence for a while.

\claim{(7.11) Lemma} If $(Y_1,\pi_1)$ and $(Y_2,\pi_2)$ are
normalizations of $X$, there is a unique analytic isomorphism
$\varphi:Y_1\longrightarrow Y_2$ such that $\pi_1=\pi_2\circ\varphi$.
\endclaim

\proof{Proof} Let $\Sigma$ be the set of singular points of $X$ and
$A_j=\pi_j^{-1}(\Sigma)$, $j=1,2$. Let $\varphi':Y_1\ssm A_1
\longrightarrow Y_2\ssm A_2$ be the analytic isomorphism $\pi_2^{-1}
\circ\pi_1$. We assert that $\varphi'$ can be extended into a map
$\varphi:Y_1\longrightarrow Y_2$. In fact, let $a\in A_1$ and $s=\pi_1(a)\in\Sigma$.
Then $\pi_2^{-1}(s)$ consists of a finite set of points $y_j\in Y_2$.
Take disjoint neighborhoods $U_j$ of $y_j$ such that $U_j$ is
an analytic subset in an open set $\Omega_j\compact\bC^N$.
Since $\pi_2$ is proper, there is a neighborhood $V$ of $s$ in $X$
such that $\pi_2^{-1}(V)\subset\bigcup U_j$ and by continuity
of $\pi_1$ a neighborhood $W$ of $a$ such that $\pi_1(W)\subset V$.
Then $\varphi'=\pi_2^{-1}\circ\pi_1$ maps $W\ssm A_1$ into $\bigcup U_j$
and can be seen as a bounded holomorphic map into $\bC^N$ through
the embeddings $U_j\subset\Omega_j\compact\bC^N$. Since $Y_1$
is normal, $\varphi'$ extends to $W$, and the extension takes values
in $\bigcup\ol U_j$ which is contained in $Y_2$ (shrink $U_j$
if necessary). Thus $\varphi'$ extends into a map $\varphi:Y_1\longrightarrow Y_2$
and similarly $\varphi^{\prime -1}$ extends into a map
$\psi:Y_2\longrightarrow Y_1$. By density of $Y_j\ssm A_j$, we have 
$\psi\circ\varphi=\Id_{Y_1}$, $\varphi\circ\psi=\Id_{Y_2}$.\qed
\endproof

\claim{(7.12) Oka normalization theorem} Let $X$ be
any complex space. Then $X$ has a normalization $(Y,\pi)$.
\endclaim

\proof{Proof} Because of the previous lemma, it suffices to prove
that any point $x\in X$ has a neighborhood $U$ such that $U$
admits a normalization; all these local normalizations will
then glue together. Hence we may suppose that $X$ is an
analytic set in an open set of $\bC^n$. Moreover,
if $(X,x)$ splits into irreducible components $(X_j,x)$ and if
$(Y_j,\pi_j)$ is a normalization of $X_j\cap U$, then the disjoint 
union $Y=\coprod Y_j$ with $\pi=\coprod\pi_j$ is easily seen to be a
normalization of $X\cap U$. We may therefore assume that $(X,x)$ is
irreducible. Let $h$ be a universal denominator in a neighborhood of $x$.
Then $\tilde\cO_{X,x}$ is isomorphic to its image $h\tilde\cO_{X,x}
\subset\cO_{X,x}$, so it is a finitely generated $\cO_{X,x}$-module.
Let $(f_1,\ldots,f_m)$ be a finite set of generators of $\cO_{X,x}$.
After shrinking $X$ again, we may assume the following two points:
\medskip
\item{$\bullet$} $X$ is an analytic set in an open set 
$\Omega\subset\bC^n$, $(X,x)$ is irreducible and $X_\reg$ 
is connected;\smallskip
\item{$\bullet$} $f_j$ is holomorphic in $X_\reg$, can be written 
$f_j=g_j/h$ on $X$ with \hbox{$g_j,h$} in $\cO_n(\Omega)$ and satisfies an 
integral equation $P_j(z\,;\,f_j(z))=0$ where $P_j(z\,;\,T)$ is a 
unitary polynomial with holomorphic coefficients on $X$.
\medskip
\noindent Set $X'=X\ssm h^{-1}(0)$. Consider the holomorphic map
$$F:X_\reg\longrightarrow\Omega\times\bC^m,~~~~
z\longmapsto\big(z,f_1(z),\ldots,f_m(z)\big)$$
and the image $Y'=F(X')$. We claim that the closure $Y$ of
$Y'$ in $\Omega\times\bC^m$ is an analytic set. In fact, the set
$$Z=\big\{(z,w)\in\Omega\times\bC^m\,;\,z\in X\,,~h(z)w_j=g_j(z)\big\}$$
is analytic and $Y'=Z\ssm\{h(z)=0\}$, so we may apply Cor.~5.4.
Observe that $Y'$ is contained in the set defined by $P_j(z\,;w_j)=0$,
thus so is its closure $Y$. The first projection 
$\Omega\times\bC^m\longrightarrow\Omega$ gives a
holomorphic map $\pi:Y\longrightarrow X$ such that $\pi\circ F=\Id$ on $X'$,
hence also on $X_\reg$. If $\Sigma=X_\sing$ and 
$A=\pi^{-1}(\Sigma)$, the restriction $\pi:Y\ssm A\longrightarrow X\ssm 
\Sigma=X_\reg$ is thus an analytic isomorphism and $F$ is its
inverse. Since $(X,x)$ is irreducible, each $f_j$ has a limit $\ell_j$ at 
$x$ by Th.~7.3 and the fiber $\pi^{-1}(x)$ is reduced to the 
single point $y=(x,\ell)$. The other fibers $\pi^{-1}(z)$ are finite 
because they are contained in the finite set of roots of the
equations $P_j(z\,;\,w_j)=0$. The same argument easily shows that 
$\pi$ is proper (use Lemma~4.10).

Next, we show that $Y$ is normal at the point $y=\pi^{-1}(x)$.
In fact, for any bounded holomorphic function $u$ on $(Y_\reg,y)$
the function $u\circ F$ is bounded and holomorphic on $(X_\reg,x)$.
Hence $u\circ F\in\tilde\cO_{X,x}=\cO_{X,x}[f_1,\ldots,f_m]$
and we can write $u\circ F(z)=Q(z\,;\,f_1(z),\ldots,f_m(z))=Q\circ F(z)$
where $Q(z\,;\,w)=\sum a_\alpha(z)w^\alpha$ is a polynomial in $w$ with
coefficients in $\cO_{X,x}$. Thus $u$ coincides with
$Q$ on $(Y_\reg,y)$, and as $Q$ is holomorphic on 
$(X,x)\times\bC^m\supset(Y,y)$,  we conclude that $u\in\cO_{Y,y}$.
Therefore $\tilde\cO_{Y,y}=\cO_{Y,y}$. 

Finally, by Th.~7.5, there is a neighborhood $V\subset Y$ of 
$y$ such that every point of $V$ is normal. As $\pi$ is proper, 
we can find a neighborhood $U$ of $x$ with $\pi^{-1}(U)\subset V$.
Then $\pi:\pi^{-1}(U)\longrightarrow U$ is the required normalization in
a neighborhood of~$x$.\qed
\endproof

The proof of Th.~7.12 shows that the
fiber $\pi^{-1}(x)$ has exactly one point $y_j$ for each irreducible
component $(X_j,x)$ of $(X,x)$. As a one-to-one proper map is a 
homeomorphism, we get in particular:

\claim{(7.13) Corollary} If $X$ is a locally irreducible complex space,
the normali\-zation $\pi:Y\longrightarrow X$ is a homeomorphism.\qed
\endclaim

\claim{(7.14) Remark} \rm In general, for any open set $U\subset X$, we 
have an isomorphism
$$\pi^\star:\tilde\cO_X(U)\buildo\simeq\over\longrightarrow
\cO_Y\big(\pi^{-1}(U)\big),\leqno(7.15)$$
whose inverse is given by the comorphism of $\pi^{-1}:X_\reg\longrightarrow Y$~;
note that $\tilde\cO_Y(U)=\cO_Y(U)$ since $Y$ is normal. Taking
the direct limit over all neighborhoods $U$ of a given point $x\in X$,
we get an isomorphism
$$\pi^\star:\tilde\cO_{X,x}\longrightarrow\bigoplus_{y_j\in\pi^{-1}(x)}\cO_{Y,y_j}.
\leqno(7.15')$$
In other words, $\tilde\cO_X$ is isomorphic to the direct image sheaf
$\pi_\star\cO_Y$, see (1.12). We will prove later on 
the deep fact that the direct image of a coherent sheaf by a proper
holomorphic map is always coherent (\cite{Grauert~1960}, see~9.?.1). Hence
$\tilde\cO_X=\pi_\star\cO_Y$ is a coherent sheaf over $\cO_X$.
\endclaim

\section{8}{Holomorphic Mappings and Extension Theorems}
\subsection{8.1}{Rank of a Holomorphic Mapping}
Our goal here is to introduce the general concept of the rank of a
holomorphic map and to relate the rank to the dimension of the fibers. 
As in the smooth case, the rank is shown to satisfy semi-continuity 
properties.

\claim{(8.1) Lemma} Let $F:X\longrightarrow Y$ be a holomorphic map from a complex 
space $X$ to a complex space $Y$.
\smallskip
\item{\rm a)} If $F$ is finite, i.e.\ proper with finite fibers,
then $\dim X\le\dim Y$.
\smallskip
\item{\rm b)} If $F$ is finite and surjective, then $\dim X=\dim Y$.
\endclaim

\proof{Proof} a) Let $x\in X$, $(X_j,x)$ an irreducible component and
$m=\dim(X_j,x)$.  If $(Y_k,y)$ are the irreducible components of $Y$ at
$y=F(x)$, then $(X_j,x)$ is contained in $\bigcup F^{-1}(Y_k)$, hence
$(X_j,x)$ is contained in one of the sets $F^{-1}(Y_k)$.  If
$p=\dim(Y_k,y)$, there is a ramified covering $\pi$ from some
neighborhood of $y$ in $Y_k$ onto a polydisk in $\Delta'\subset\bC^p$.  
Replacing $X$ by some neighborhood of $x$ in $X_j$ and $F$ by
the finite map $\pi\circ F_{\restriction X_j}:X_j\longrightarrow\Delta'$, we may 
suppose that $Y=\Delta'$ and that $X$ is irreducible, $\dim X=m$.  Let 
$r=\hbox{\rm rank}\,dF_{x_0}$ be the maximum of the rank of the differential 
of $F$ on $X_\reg$.  Then $r\le\min\{m,p\}$ and the rank of $dF$ is
constant equal to $r$ on a neighborhood $U$ of $x_0$.  The constant
rank theorem implies that the fibers $F^{-1}(y)\cap U$ are
$(m-r)$-dimensional submanifolds, hence $m-r=0$ and $m=r\le p$.
\medskip
\noindent{\rm b)} We only have to show that $\dim X\ge\dim Y$. Fix a 
regular point $y\in Y$ of maximal dimension. By taking
the restriction $F:F^{-1}(U)\longrightarrow U$ to a small neighborhood $U$
of $y$, we may assume that $Y$ is an open subset of $\bC^p$.
If $\dim X<\dim Y$, then $X$ is a union of analytic manifolds
of dimension $<\dim Y$ and Sard's theorem implies that $F(X)$ has
zero Lebesgue measure in $Y$, a contradiction.\qed
\endproof

\claim{(8.2) Proposition} For any holomorphic map $F:X\longrightarrow Y$,
the fiber dimension $\dim\big(F^{-1}(F(x)),x\big)$ is an upper
semi-continuous function of $x$.
\endclaim

\proof{Proof} Without loss of generality, we may suppose that $X$ is an
analytic set in $\Omega\subset\bC^n$, that $F(X)$ is contained in a
small neighborhood of $F(x)$ in $Y$ which is embedded in $\bC^N$, and
that $x=0$, $F(x)=0$.  Set $A=F^{-1}(0)$ and $s=\dim(A,0)$.  We can
find a linear form $\xi_1$ on $\bC^n$ such that
$\dim(A\cap\xi_1^{-1}(0),0)=s-1$~; in fact we need only select a point
$x_j\ne 0$ on each irreducible component $(A_j,0)$ of $(A,0)$ and take
$\xi_1(x_j)\ne 0$.  By induction, we can find linearly independent forms
$\xi_1,\ldots,\xi_s$ on $\bC^n$ such that
$$\dim\big(A\cap\xi_1^{-1}(0)\cap\ldots\cap\xi_j^{-1}(0),0\big)=s-j$$
for all $j=1,\ldots,s$~; in particular $0$ is an isolated point in the
intersection when $j=s$.  After a change of coordinates, we may suppose
that $\xi_j(z)=z_j$.  Fix $r''$ so small that the ball $\ol B''
\subset\bC^{n-s}$ of center $0$ and radius $r''$ satisfies
$A\cap(\{0\}\times\ol B'')=\{0\}$.  Then $A$ is disjoint from the
compact set $\{0\}\times\partial B''$, so there exists a small ball
$B'\subset\bC^s$ of center $0$ such that $A\cap(\ol B'\times\partial B'')=
\emptyset$, i.e.\ $F$ does not vanish on the compact set
$K=X\cap(\ol B'\times\partial B'')$.  Set $\varepsilon=\min_K|F|$. 
Then for $|y|<\varepsilon$ the fiber $F^{-1}(y)$ does not intersect
$\ol B'\times\partial B''$.  This implies that the projection map
$\pi:F^{-1}(y)\cap(B'\times B'')\longrightarrow B'$ is proper.  The fibers of $\pi$ are
then compact analytic subsets of $B''$, so they are finite by~5.9. 
Lemma~8.1~a) implies 
$$\dim F^{-1}(y)\cap(B'\times B'')\le\dim B'=s=\dim(A,0)=
\dim(F^{-1}(0),0).\eqno\square$$
\endproof

Let $X$ be a pure dimensional complex space and $F:X\longrightarrow Y$ a
holomorphic map. For any point $x\in X$, we define the 
{\it rank of $F$ at $x$} by 
$$\rho_F(x)=\dim(X,x)-\dim\big(F^{-1}(F(x)),x\big).\leqno(8.3)$$
By the above proposition, $\rho_F$ is a lower semi-continuous
function on $X$. In particular, if $\rho_F$ is maximum at some
point $x_0$, it must be constant in a neighborhood of $x_0$.
The maximum $\ol\rho(F)=\max_X\rho_F$ is thus attained on 
$X_\reg$ or on any dense open subset $X'\subset X_\reg$. 
If $X$ is not pure dimensional, we define $\ol\rho(F)=\max_\alpha
\ol\rho(F_{\restriction X_\alpha})$ where $(X_\alpha)$ are the
irreducible components of $X$. For a map $F:X\longrightarrow\bC^N$, the 
constant rank theorem implies that $\ol\rho(F)$ is equal to the 
maximum of the rank of the jacobian matrix $dF$ at 
points of $X_\reg$ (or of $X'$).

\claim{(8.4) Proposition} If $F:X\longrightarrow Y$ is a holomorphic map and $Z$ 
an analytic subset of $X$, then $\ol\rho(F_{\restriction Z})\le
\ol\rho(F)$.
\endclaim

\proof{Proof} Since each irreducible component of $Z$ is contained
in an irreducible component of $X$, we may assume $X$ irreducible.
Let $\pi:\tilde X\longrightarrow X$ be the normalization of $X$ and 
$\tilde Z=\pi^{-1}(Z)$. Since $\pi$ is finite and surjective,
the fiber of $F\circ\pi$ at point $x$ has the same dimension than the 
fiber of $F$ at $\pi(x)$ by Lemma~8.1~b). Therefore 
$\ol\rho(F\circ\pi)=\ol\rho(F)$
and $\ol\rho(F\circ\pi_{\restriction\tilde Z})=\ol\rho(F_{\restriction Z})$,
so we may assume $X$ normal. By induction on $\dim X$,
we may also suppose that $Z$ has pure codimension 1 in $X$ (every point
of $Z$ has a neighborhood $V\subset X$ such that $Z\cap V$ is contained 
in a pure one codimensional analytic subset of $V$). But
then $Z_\reg\cap X_\reg$ is dense in $Z_\reg$
because $\codim X_\sing\ge 2$. Thus we are reduced to the 
case when $X$ is a manifold and $Z$ a submanifold, and this case 
is clear if we consider the rank of the jacobian matrix.\qed
\endproof

\claim{(8.5) Theorem} Let $F:X\longrightarrow Y$ be a holomorphic map.
If $Y$ is pure dimensional and $\ol\rho(F)<\dim Y$, then
$F(X)$ has empty interior in $Y$.
\endclaim

\proof{Proof} Taking the restriction of $F$ to $F^{-1}(Y_\reg)$, we may
assume that $Y$ is a manifold. Since $X$ is a countable union of 
compact sets, so is $F(X)$, and Baire's theorem shows that the 
result is local for $X$. By Prop.~8.4 and an induction on 
$\dim X$, $F(X_\sing)$ has empty interior in $Y$.
The set $Z\subset X_\reg$ of points where the jacobian matrix
of $F$ has rank $<\ol\rho(F)$ is an analytic subset
hence, by induction again, $F(Z)$ has empty interior. The
constant rank theorem finally shows that every point $x\in X_\reg
\ssm Z$ has a neighborhood $V$ such that $F(V)$ is a
submanifold of dimension $\ol\rho(F)$ in $Y$, thus
$F(V)$ has empty interior and Baire's theorem completes the proof.\qed
\endproof

\claim{(8.6) Corollary} Let $F:X\longrightarrow Y$ be a surjective holomorphic map.
Then $\dim Y=\ol\rho(F)$.
\endclaim

\proof{Proof} By the remark before Prop.~8.4, there is a regular point
$x_0\in X$ such that the jacobian matrix of $F$ has rank
$\ol\rho(F)$. Hence, by the constant rank theorem $\dim Y\ge\ol\rho(F)$.
Conversely, let $Y_\alpha$ be an irreducible component of $Y$
of dimension equal to $\dim Y$, and $Z=F^{-1}(Y_\alpha)\subset X$.
Then $F(Z)=Y_\alpha$ and Th.~8.5 implies
$\ol\rho(F)\ge\ol\rho(F_{\restriction Z})\ge\dim Y_\alpha$.\qed
\endproof

\subsection{8.2}{Remmert and Remmert-Stein Theorems}
We are now ready to prove two important results:
the extension theorem for analytic subsets due to \cite{Remmert and 
Stein 1953} and the theorem of \cite{Remmert 1956, 1957} which asserts
that the image of a complex space under a proper holomorphic map is an
analytic set.  These will be obtained by a simultaneous induction on
the dimension.

\claim{(8.7) Remmert-Stein theorem} Let $X$ be a complex space, 
$A$ an analytic subset of $X$ and $Z$ an analytic subset of 
$X\ssm A$. Suppose that there is an integer $p\ge 0$ such 
that $\dim A\le p$, while $\dim(Z,x)>p$ for all $x\in Z$. 
Then the closure $\ol Z$ of $Z$ in $X$ is an analytic subset.
\endclaim

\claim{(8.8) Remmert's proper mapping theorem} Let 
$F:X\longrightarrow Y$ be a proper holomorphic map.  Then $F(X)$ is an analytic
subset of $Y$.
\endclaim

\proof{Proof} We let $(8.7_m)$ denote statement (8.7) for $\dim Z\le m$
and $(8.8_m)$ denote statement (8.8) for $\dim X\le m$. We proceed
by induction on $m$ in two steps:\medskip
\noindent{\it Step 1.} $(8.7_m)$ and $(8.8_{m-1})$ imply 
$(8.8_m)$.\\
\noindent{\it Step 2.} $(8.8_{m-1})$ implies $(8.7_m)$.
\medskip
\noindent As $(8.8_m)$ is obvious for $m=0$, our statements will then be valid
for all $m$, i.e.\ for all complex spaces of bounded dimension. However,
Th.~8.7 is local on $X$ and Th.~8.8 is local on $Y$,
so the general case is immediately reduced to the finite dimensional case.
\medskip
\noindent{\it Proof of step 1.} The analyticity of $F(X)$ is a
local question in $Y$. Since $F:F^{-1}(U)\longrightarrow U$ is proper for any
open set $U\subset Y$ and $F^{-1}(U)\compact X$ if $U\compact Y$, 
we may suppose that $Y$ is embedded in an open set $\Omega\subset\bC^n$ 
and that $X$ only has finitely many irreducible components $X_\alpha$. 
Then we have $F(X)=\bigcup F(X_\alpha)$ and we are reduced to the case when 
$X$ is irreducible, $\dim X=m$ and $Y=\Omega$. 

First assume that $X$ is a manifold and that the rank of $dF$ is constant.
The constant rank theorem implies that every point in $X$ has a neighborhood
$V$ such that $F(V)$ is a closed submanifold in a neighborhood
$W$ of $F(x)$ in $Y$. For any point $y\in Y$, the fiber $F^{-1}(y)$ 
can be covered by finitely many neighborhoods $V_j$ of points
$x_j\in F^{-1}(y)$ such that $F(V_j)$ is a closed submanifold in a 
neighborhood $W_j$ of $y$. Then there is a neighborhood of $y$
$W\subset\bigcap W_j$ such that $F^{-1}(W)\subset\bigcup V_j$,
so $F(X)\cap W=\bigcup F(V_j)\cap W$ is a finite union of closed
submanifolds in $W$ and $F(X)$ is analytic in $Y$. 

Now suppose that $X$ is a manifold, set $r=\ol\rho(F)$ and
let $Z\subset X$ be the analytic subset of points $x$ where the
rank of $dF_x$ is $<r$. Since $\dim Z<m=\dim X$, the hypothesis
$(8.8_{m-1})$ shows that $F(Z)$ is analytic. We
have $\dim F(Z)=\ol\rho(F_{\restriction Z})<r$. If $F(Z)=F(X)$, then
$F(X)$ is analytic. Otherwise $A=F^{-1}\big(F(Z)\big)$ is a 
proper analytic subset of $X$, $dF$ has constant rank on $X\ssm A
\subset X\ssm Z$ and the morphism $F:X\ssm A\longrightarrow Y\ssm F(Z)$ 
is proper. Hence the image $F(X\ssm A)$ is analytic in $Y\ssm F(Z)$.
Since $\dim F(X\ssm A)=r\le m$ and $\dim F(Z)<r$, hypothesis $(8.7_m)$
implies that $F(X)=\ol{F(X\ssm A)}$ is analytic in $Y$. When $X$ 
is not a manifold, we apply the same reasoning with $Z=X_\sing$ in order
to be reduced to the case of $F:X\ssm A\longrightarrow Y\ssm F(Z)$
where $X\ssm A$ is a manifold.\qed
\endproof

\noindent{\it Proof of step 2.} 
Since Th.~8.7 is local on $X$, we may suppose that $X$ is an 
open set $\Omega\subset\bC^n$. Then we use induction on $p$ to 
reduce the situation to the case when 
$A$ is a $p$-dimensional submanifold (if this case is taken for granted,
the closure of $Z$ in $\Omega\ssm A_\sing$ is analytic and we 
conclude by the induction hypothesis). By a local analytic change of
coordinates, we may assume that $0\in A$ and that $A=\Omega\cap L$ 
where $L$ is a vector subspace of $\bC^n$ of dimension $p$.
By writing $Z=\bigcup_{p<s\le m}Z_s$ 
where $Z_s$ is an analytic subset of $\Omega\ssm Y$ of pure 
dimension $s$, we may suppose that $Z$ has pure dimension~$s$,
$p<s\le m$. We are going to show that $\ol Z$ is analytic in a 
neighborhood~of~$0$.

Let $\xi_1$ be a linear form on $\bC^n$ which is not identically zero
on $L$ nor on any irreducible component of $Z$ (just pick a point
$x_\nu$ on each component and take $\xi_1(x_\nu)\ne 0$ for all $\nu$). 
Then $\dim L\cap \xi_1^{-1}(0)=p-1$ and the analytic set 
$Z\cap \xi_1^{-1}(0)$ has pure dimension $s-1$. By induction, 
there exist linearly independent forms $\xi_1,\ldots,\xi_s$ such that
$$\leqalignno{
\dim L\cap \xi_1^{-1}(0)\cap\ldots\cap \xi_j^{-1}(0)&=p-j,~~~~
1\le j\le p,\cr
\dim Z\cap \xi_1^{-1}(0)\cap\ldots\cap \xi_j^{-1}(0)&=s-j,~~~~
1\le j\le s.&(8.9)\cr}$$

\noindent By adding a suitable linear combination of $\xi_1,\ldots,\xi_p$ 
to each $\xi_j$, $p<j\le s$, we may take $\xi_{j\restriction L}=0$
for $p<j\le s$. After a linear change of coordinates, we may suppose that
$\xi_j(z)=z_j$, $L=\bC^p\times\{0\}$ and $A=\Omega\cap(\bC^p\times\{0\})$.
Let $\xi=(\xi_1,\ldots,\xi_s):\bC^n\longrightarrow\bC^s$ be the projection onto the
first $s$ variables. As $Z$ is closed in $\Omega\ssm A$, $Z\cup A$ 
is closed in $\Omega$. Moreover, our construction gives
$(Z\cup A)\cap\xi^{-1}(0)=\big(Z\cap\xi^{-1}(0)\big)\cup\{0\}$ 
and the case $j=s$ of (8.9) shows
that $Z\cap\xi^{-1}(0)$ is a locally finite sequence in 
$\Omega\cap(\{0\}\times\bC^{n-s})\ssm\{0\}$. Therefore, we can 
find a small ball $\ol B''$ of center $0$ in $\bC^{n-s}$
such that $Z\cap(\{0\}\times\partial B'')=\emptyset$.
As $\{0\}\times\partial B''$ is compact and disjoint from the closed
set $Z\cup A$, there is a small ball $B'$ of center $0$ in $\bC^s$
such that $(Z\cup A)\cap(\ol B'\times\partial B'')=\emptyset$. This
implies that the projection $\xi:(Z\cup A)\cap(B'\times B'')\longrightarrow B'$
is proper. Set $A'=B'\cap(\bC^p\times\{0\})$. 

\vbox{%
\input epsfiles/fig_2_3.tex
\vskip1mm
\centerline{{\bf Fig.~II-3} Projection $\pi:Z\cap((B'\ssm A')\times B'')
\longrightarrow B'\ssm A'$.}}
\vskip5mm

\noindent
Then the restriction
$$\pi=\xi:Z\cap(B'\times B'')\ssm (A'\times B'')\longrightarrow B'\ssm A'$$
is proper, and $Z\cap(B'\times B'')$ is analytic in $(B'\times B'')\ssm
A$, so $\pi$ has finite fibers by Th.~5.9. By definition of the rank
we have $\ol\rho(\pi)=s$. Let $S_1=Z_\sing\cap\pi^{-1}(B'\ssm A')$ and
$S'_1=\pi(S_1)$~; further, let $S_2$ be the set of points $x\in
Z\cap\pi^{-1}\big(B'\ssm(A'\cup S'_1)\big)\subset Z_\reg$ such that
$d\pi_x$ has rank $<s$ and $S'_2=\pi(S_2)$. We have $\dim S_j\le s-1\le m-1$.
Hypothesis $(8.8)_{m-1}$ implies that $S'_1$ is analytic in $B'\ssm A'$
and that $S'_2$ is analytic in $B'\ssm(A'\cup S'_1)$. By Remark~4.2,
$B'\ssm(A'\cup S'_1\cup S'_2)$ is connected and every bounded
holomorphic function on this set extends to $B'$. As $\pi$ is a (non ramified)
covering over $B'\ssm(A'\cup S'_1\cup S'_2)$, the sheet
number is a constant $q$.

Let $\lambda(z)=\sum_{j>s}\lambda_jz_j$ be a 
linear form on $\bC^n$ in the coordinates of index $j>s$. For
$z'\in B'\ssm(A'\cup S'_1\cup S'_2)$, we let $\sigma_j(z')$ be the
elementary symmetric functions in the $q$ complex numbers $\lambda(z)$ 
corresponding to $z\in\pi^{-1}(z')$. Then these functions can be 
extended as bounded holomorphic functions on $B'$ and we get a polynomial
$P_\lambda(z'\,;\,T)$ such that $P_\lambda\big(z'\,;\,\lambda(z'')\big)$
vanishes identically on $Z\ssm\pi^{-1}(A'\cup S'_1\cup S'_2)$. 
Since $\pi$ is finite, $Z\cap\pi^{-1}(A'\cup S'_1\cup S'_2)$
is a union of three (non necessarily closed) analytic subsets of dimension 
$\le s-1$, thus has empty interior in $Z$. It follows that 
the closure $\ol Z\cap(B'\times B'')$ is contained in the analytic set 
$W\subset B'\times B''$ equal to the common zero set of all functions 
$P_\lambda\big(z'\,;\,\lambda(z'')\big)$. Moreover, by construction,
$$Z\ssm\pi^{-1}(A'\cup S'_1\cup S'_2)=
W\ssm\pi^{-1}(A'\cup S'_1\cup S'_2).$$
As in the proof of Cor.~5.4, we easily conclude that
$\ol Z\cap(B'\times B'')$ is equal to the union of all irreducible 
components of $W$ that are not contained in $\pi^{-1}(A'\cup S'_1\cup S'_2)$.
Hence $\ol Z$ is analytic.\qed\medskip

Finally, we give two interesting applications of the Remmert-Stein theorem.
We assume here that the reader knows what is the complex projective
space $\bP^n$. For more details, see Sect. 5.15.

\claim{(8.10) Chow's theorem {\rm(\cite{Chow 1949})}} Let 
$A$ be an analytic subset of the complex projective space
$\bP^n$.  Then $A$ is algebraic, i.e.\ $A$ is the common zero set of
finitely many homogeneous polynomials $P_j(z_0,\ldots,z_n)$, $1\le j\le N$.
\endclaim

\proof{Proof} Let $\pi:\bC^{n+1}\ssm\{0\}\longrightarrow\bP^n$ be the natural
projection and $Z=\pi^{-1}(A)$. Then $Z$ is an analytic subset
of $\bC^{n+1}\ssm\{0\}$ which is invariant by homotheties
and $\dim Z=\dim A+1\ge 1$. The Remmert-Stein theorem implies
that $\ol Z=Z\cup\{0\}$ is an analytic subset of $\bC^{n+1}$. Let
$f_1,\ldots,f_N$ be holomorphic functions on a small polydisk
$\Delta\subset\bC^{n+1}$ of center $0$ such that
$\ol Z\cap\Delta=\bigcap f_j^{-1}(0)$. The Taylor series
at $0$ gives an expansion $f_j=\sum_{k=0}^{+\infty}P_{j,k}$ where
$P_{j,k}$ is a homogeneous polynomial of degree $k$.
We claim that $\ol Z$ coincides with the common zero $W$ set of the
polynomials $P_{j,k}$. In fact, we clearly have $W\cap\Delta
\subset\bigcap f_j^{-1}(0)=\ol Z\cap\Delta$. Conversely, for 
$z\in\ol Z\cap\Delta$, the invariance of $Z$ by homotheties shows that
$f_j(tz)=\sum P_{j,k}(z)t^k$ vanishes for every complex number $t$
of modulus $<1$, so all coefficients $P_{j,k}(z)$ vanish
and $z\in W\cap\Delta$. By homogeneity $\ol Z=W$~; since
$\bC[z_0,\ldots,z_n]$ is Noetherian, $W$ can be defined by finitely
many polynomial equations.\qed
\endproof

\claim{(8.11) E.E.\ Levi's continuation theorem} Let $X$ 
be a normal complex space and $A$ an analytic subset 
such that $\dim(A,x)\le\dim(X,x)-2$ for all $x\in A$. Then every
meromorphic function on $X\ssm A$ has a meromorphic
extension to $X$.
\endclaim

\proof{Proof} We may suppose $X$ irreducible, $\dim X=n$.
Let $f$ be a meromorphic function on $X\ssm A$. By Th.~6.13,
the pole set $P_f$ has pure dimension $(n-1)$, so
the Remmert-Stein theorem implies that $\ol P_f$ is analytic in $X$.
Fix a point $x\in A$. There is a connected neighborhood $V$ of $x$
and a non zero holomorphic function $h\in\cO_X(V)$ such that
$\ol P_f\cap V$ has finitely many irreducible components $\ol P_{f,j}$ and
$\ol P_f\cap V\subset h^{-1}(0)$. Select a point $x_j$ in  
$\ol P_{f,j}\ssm(X_\sing\cup(\ol P_f)_\sing\cup A)$.
As $x_j$ is a regular point on $X$ and on $\ol P_f$, there is a
local coordinate $z_{1,j}$ at $x_j$ defining an equation of 
$\ol P_{f,j}$, such that $z_{1,j}^{m_j}f\in\cO_{X,x_j}$
for some integer $m_j$. Since $h$ vanishes along $P_f$, we have 
$h^{m_j}f\in\cO_{X,x}$. Thus, for $m=\max\{m_j\}$, the pole set 
$P_g$ of $g=h^m f$ in $V\ssm A$ does not contain $x_j$.
As $P_g$ is $(n-1)$-dimensional and contained in
$P_f\cap V$, it is a union of irreducible components 
$\ol P_{f,j}\ssm A$. Hence $P_g$ must be empty and
$g$ is holomorphic on $V\ssm A$. By Cor.~7.8,
$g$ has an extension to a holomorphic function $\tilde g$ on $V$. Then
$\tilde g/h^m$ is the required meromorphic extension of $f$ on~$V$.\qed
\endproof

\section{9}{Complex Analytic Schemes}
Our goal is to introduce a generalization of the notion of complex space
given in Def.~5.2.  A complex space is a space locally isomorphic to
an analytic set $A$ in an open subset $\Omega\subset\bC^n$, together
with the sheaf of rings $\cO_A=(\cO_\Omega/\cI_A)_{\restriction A}$.
Our desire is to enrich the structure sheaf $\cO_A$ by replacing
$\cI_A$ with a possibly smaller ideal $\cJ$ defining the same zero
variety $V(\cJ)=A$. In this way holomorphic functions are described not
merely by their values on $A$, but also possibly by some ``transversal
derivatives'' along $A$. 

\subsection{9.1}{Ringed Spaces}
We start by an abstract notion of ringed space on an arbitrary topological
space.

\claim{(9.1) Definition} A ringed space is a pair $(X,\cR_X)$ consisting 
of a topolo\-gical space $X$ and of a sheaf of rings $\cR_X$ on $X$, called the
structure sheaf. A~morphism
$$F:(X,\cR_X)\longrightarrow(Y,\cR_Y)$$
of ringed spaces is a pair $(f,F^\star)$ where $f:X\longrightarrow Y$ is a
continuous map and
$$F^\star~:~~f^{-1}\cR_Y\longrightarrow\cR_X,~~~~F^\star_x~:~~(\cR_Y)_{f(x)}
\longrightarrow(\cR_X)_x$$
a homomorphism of sheaves of rings on $X$, called the comorphism of $F$.
\endclaim

If $F:(X,\cR_X)\longrightarrow(Y,\cR_Y)$ and $G:(Y,\cR_Y)\longrightarrow(Z,\cR_Z)$ are
morphisms of ringed spaces, the composite $G\circ F$ is the
pair consisting of the map $g\circ f:X\longrightarrow Z$ and of the comorphism
$(G\circ F)^\star=F^\star\circ f^{-1}G^\star\,$:
$$\cmalign{
\hfill&F^\star\circ f^{-1}G^\star~:~~&f^{-1}g^{-1}\cR_Z~\,&\kern-1.9mm
\buildo f^{-1}G^\star\over\llra f^{-1}\cR_Y
&\kern-1.9mm\buildo F^\star\over{\relbar\mkern-4mu\longrightarrow}\cR_X,\cr
&F^\star_x\circ G^\star_{f(x)}~:~~~&(\cR_Z)_{g\circ f(x)}~\,&{\relbar\mkern-4mu
\relbar\mkern-4mu\longrightarrow}~(\cR_Y)_{f(x)}~&{\relbar\mkern-4mu\longrightarrow}~(\cR_X)_x.\cr}
\leqno(9.2)$$

\subsection{9.2}{Definition of Complex Analytic Schemes}
We begin by a description of what will be the local model of an
analytic scheme. Let $\Omega\subset\bC^n$ be an open subset,
$\cJ\subset\cO_\Omega$ a coherent sheaf of ideals and $A=V(\cJ)$
the analytic set in $\Omega$ defined locally as the zero set of
a system of generators of $\cJ$. By Hilbert's Nullstellensatz~4.22
we have $\cI_A=\sqrt\cJ$, but $\cI_A$ differs in general from $\cJ$. 
The sheaf of rings $\cO_\Omega/\cJ$ is supported on $A$, 
i.e.\ $(\cO_\Omega/\cJ)_x=0$ if $x\notin A$. Ringed spaces of the type
$(A,\cO_\Omega/\cJ)$ will be used as the local models of analytic 
schemes.

\claim{(9.3) Definition} A morphism 
$$F=(f,F^\star):(A,\cO_\Omega/\cJ_{\restriction A})\longrightarrow(A',\cO_{\Omega'}/\cJ'
_{\restriction A'})$$
is said to be analytic if for every point $x\in A$ there exists a 
neighborhood $W_x$ of $x$ in $\Omega$ and a holomorphic function
$\Phi:W_x\longrightarrow\Omega'$ such that $f_{\restriction A\cap W_x}=
\Phi_{\restriction A\cap W_x}$ and such that the comorphism
$$F^\star_x:(\cO_{\Omega'}/\cJ')_{f(x)}\longrightarrow(\cO_\Omega/\cJ)_x$$
is induced by $\Phi^\star:\cO_{\Omega',f(x)}\ni u\longmapsto u\circ\Phi
\in\cO_{\Omega,x}$ with $\Phi^\star\cJ'\subset\cJ$.
\endclaim

\claim{(9.4) Example} \rm Take $\Omega=\bC^n$ and $\cJ=(z_n^2)$. Then $A$
is the hyperplane $\bC^{n-1}\times\{0\}$, and the sheaf
$\cO_{\bC^n}/\cJ$ can be identified with the sheaf of rings of functions
$u+z_nu'$, $u,u'\in\cO_{\bC^{n-1}}$, with the relation $z_n^2=0$.
In particular, $z_n$ is a nilpotent element of $\cO_{\bC^n}/\cJ$.
A morphism $F$ of $(A,\cO_{\bC^n}/\cJ)$ into itself is induced
(at least locally) by a holomorphic map $\Phi=(\wt\Phi,\Phi_n)$
defined on a neighborhood of $A$ in $\bC^n$ with values in $\bC^n$, such that
$\Phi(A)\subset A$, i.e.\ $\Phi_{n\restriction A}=0$. We see that $F$
is completely determined by the data
$$\cmalign{
\hfill f(z_1,\ldots,z_{n-1})&=\wt\Phi(z_1,\ldots,z_{n-1},0),~~~~\hfill
f~&:~~\bC^{n-1}\longrightarrow\bC^{n-1},\cr
f'(z_1,\ldots,z_{n-1})&=\displaystyle{\partial\Phi\over\partial z_n}
(z_1,\ldots,z_{n-1},0),~~~~f'~&:~~\bC^{n-1}\longrightarrow\bC^n,\cr}$$
which can be chosen arbitrarily.
\endclaim

\claim{(9.5) Definition} A complex analytic scheme is a ringed space
$(X,\cO_X)$ over a separable Hausdorff topological space $X$,
satisfying the following property: there exist an open covering
$(U_\lambda)$ of $X$ and isomorphisms of ringed spaces 
$$G_\lambda:(U_\lambda,\cO_{X\restriction U_\lambda})
\longrightarrow(A_\lambda,\cO_{\Omega_\lambda}/\cJ_{\lambda\,\restriction A_\lambda})$$
where $A_\lambda$ is the zero set of a coherent sheaf of ideals $\cJ_\lambda$
on an open subset $\Omega_\lambda\subset\bC^{N_\lambda}$,
such that every transition morphism $G_\lambda\circ G_\mu^{-1}$
is a holomorphic isomorphism from $g_\mu(U_\lambda\cap U_\mu)\subset A_\mu$
onto $g_\lambda(U_\lambda\cap U_\mu)\subset A_\lambda$, equipped with the
respective structure sheaves $\cO_{\Omega_\mu}/\cJ_{\mu\,\restriction A_\mu}$,
$\cO_{\Omega_\lambda}/\cJ_{\lambda\,\restriction A_\lambda}$.
\endclaim

We shall often consider the maps $G_\lambda$ as identifications and
write simply $U_\lambda=A_\lambda$.    
A morphism $F:(X,\cO_X)\longrightarrow(Y,\cO_Y)$ of analytic schemes obtained by
gluing patches $(A_\lambda,\cO_{\Omega_\lambda}/\cJ_{\lambda\,
\restriction A_\lambda})$ and $(A'_\mu,\cO_{\Omega'_\mu}/\cJ'_{\mu\,A'_\mu})$,
respectively, is a morphism $F$ of ringed spaces such that for each pair
$(\lambda,\mu)$, the restriction of $F$ from $A_\lambda\cap
f^{-1}(A'_\mu)\subset X$ to $A'_\mu\subset Y$ is holomorphic in the
sense of Def.~9.3.

\subsection{9.3}{Nilpotent Elements and Reduced Schemes}
Let $(X,\cO_X)$ be an analytic scheme. The set of {\it nilpotent
elements} is the sheaf of ideals of $\cO_X$ defined by
$$\cN_X=\{u\in\cO_X\,;\,u^k=0~~\hbox{\rm for~some}~\,k\in\bN\}.\leqno(9.6)$$
Locally, we have $\cO_{X\restriction A_\lambda}=
(\cO_{\Omega_\lambda}/\cJ_\lambda)_{\restriction A_\lambda}$, thus
$$\leqalignno{
&\cN_{X\restriction A_\lambda}=(\sqrt{\cJ_\lambda}/\cJ_\lambda
)_{\restriction A_\lambda},&(9.7)\cr
&(\cO_X/\cN_X)_{\restriction A_\lambda}\simeq(\cO_{\Omega_\lambda}/
\sqrt{\cJ_\lambda})_{\restriction A_\lambda}=
(\cO_{\Omega_\lambda}/\cI_{A_\lambda})_{\restriction A_\lambda}=
\cO_{A_\lambda}.&(9.8)\cr}$$
The scheme $(X,\cO_X)$ is said to be {\it reduced} if $\cN_X=0$. The
associated ringed space $(X,\cO_X/\cN_X)$ is reduced by construction;
it is called the {\it reduced scheme} of $(X,\cO_X)$. We shall often denote 
the original scheme by the letter $X$ merely, the associated reduced
scheme by $X_\red$, and let $\cO_{X,\red}=\cO_X/\cN_X$.
There is a canonical morphism $X_\red\to X$ whose comorphism is
the reduction morphism
$$\cO_X(U)\longrightarrow\cO_{X,\red}(U)=(\cO_X/\cN_X)(U),\qquad
\hbox{$\forall U$ open set in $X$}.\leqno(9.9)$$
By~(9.8), the notion of reduced scheme is equivalent to the notion of
complex space introduced in Def.~5.2. It is easy to see that a morphism
$F$ of reduced schemes $X,Y$ is completely determined by the
set-theoretic map $f:X\longrightarrow Y$.

\subsection{9.4}{Coherent Sheaves on Analytic Schemes}
If $(X,\cO_X)$ is an analytic scheme, a sheaf $\cS$ of $\cO_X$-modules
is said to be {\it coherent} if it satisfies the same properties as
those already stated when $X$ is a manifold:
\smallskip
\noindent{$(9.10)$}~~$\cS$ is locally finitely generated over $\cO_X\,$;
\Fbreak
\noindent{$(9.10')$}~for any open set $U\subset X$ and any sections
$G_1,\ldots,G_q\in\cS(U)$, the
\phantom{$(9.10')$~}relation sheaf $\cR(G_1,\ldots,G_q)\subset
\cO_{X\restriction U}^{\oplus q}$ is locally finitely generated.
\smallskip
\noindent Locally, we have 
$\cO_{X\restriction A_\lambda}=\cO_{\Omega_\lambda}/\cJ_\lambda$,
so if $i_\lambda:A_\lambda\to\Omega_\lambda$ is the injection, the
direct image $\cS_\lambda=(i_\lambda)_\star(\cS_{\restriction A_\lambda})$ is
a module over $\cO_{\Omega_\lambda}$ such that $\cJ_\lambda.\cS_\lambda=0$. 
It is clear that $\cS_{\restriction\Omega_\lambda}$ is coherent if and only if
$\cS_\lambda$ is coherent as a module over $\cO_{\Omega_\lambda}$. 
It follows immediately that the Oka theorem and its consequences 3.16--20 
are still valid over analytic schemes.

\subsection{9.5}{Subschemes}
Let $X$ be an analytic scheme and $\cG$ a coherent sheaf of ideals in
$\cO_X$. The image of $\cG$ in $\cO_{X,\red}$ is a
coherent sheaf of ideals, and its zero set $Y$ is clearly an
analytic subset of $X_\red$. We can make $Y$ into a scheme by
introducing the structure sheaf
$$\cO_Y=(\cO_X/\cG)_{\restriction Y},\leqno(9.11)$$
and we have a scheme morphism $F:(Y,\cO_Y)\longrightarrow(X,\cO_X)$ such that $f$ is the 
inclusion and 
$F^\star:f^{-1}\cO_X\longrightarrow\cO_Y$ the obvious map of $\cO_{X\restriction Y}$ onto
its quotient $\cO_Y$. The scheme $(Y,\cO_Y)$ will be denoted $V(\cG)$.
When the analytic set $Y$ is given, the structure sheaf of $V(\cG)$ depends
of course on the choice of the equations of $Y$ in the ideal $\cG\,$;
in general $\cO_Y$ has nilpotent elements.

\subsection{9.6}{Inverse Images of Coherent Sheaves}
Let $F:(X,\cO_X)\longrightarrow(Y,\cO_Y)$ be a morphism of analytic schemes and $\cS$
a coherent sheaf over~$Y$. The sheaf theoretic inverse image $f^{-1}\cS$,
whose stalks are $(f^{-1}\cS)_x=\cS_{f(x)}$, is a sheaf of modules over
$f^{-1}\cO_Y$. We define the {\it analytic inverse image} $F^\star\cS$ by
$$F^\star\cS=\cO_X\otimes_{f^{-1}\cO_Y}f^{-1}\cS,~~~~
(F^\star\cS)_x=\cO_{X,x}\otimes_{\cO_{Y,f(x)}}\cS_{f(x)}.\leqno(9.12)$$
Here the tensor product is taken with respect to the comorphism
$F^\star:f^{-1}\cO_Y\to\cO_X$, which yields a ring morphism
$\cO_{Y,f(x)}\to\cO_{X,x}$. If $\cS$ is given over $U\subset Y$ by a
local presentation
$$\cO_{Y\restriction U}^{\oplus p}\buildo A\over\longrightarrow\cO_{Y\restriction U}^{\oplus q}\longrightarrow
\cS_{\restriction U}\longrightarrow 0$$
where $A$ is a $(q\times p)$-matrix with coefficients in $\cO_Y(U)$,
our definition shows that $F^\star\cS$ is a coherent sheaf over $\cO_X$,
given over $f^{-1}(U)$ by the local presentation
$$\cO_{X\restriction f^{-1}(U)}^{\oplus p}\buildo{F^\star A}\over{\larex 24 }
\cO_{X\restriction f^{-1}(U)}^{\oplus q}\longrightarrow F^\star\cS_{\restriction f^{-1}(U)}
\longrightarrow 0.\leqno(9.13)$$

\subsection{9.7}{Products of Analytic Schemes}
Let $(X,\cO_X)$ and $(Y,\cO_Y)$ be analytic schemes, and let
$(A_\lambda,\cO_{\Omega_\lambda}/\cJ_\lambda)$,
$(B_\mu,\cO_{\Omega'_\mu}/\cJ'_\mu)$ be local models of $X$, $Y$,
respectively. The {\it product scheme} $(X\times Y,\cO_{X\times Y})$
is obtained by gluing the open patches
$$\Big(A_\lambda\times B_\mu~,~\cO_{\Omega_\lambda\times\Omega'_\mu}\big/
\big({\rm pr}_1^{-1}\cJ_\lambda+{\rm pr}_2^{-1}\cJ'_\mu\big)
\cO_{\Omega_\lambda\times\Omega'_\mu}\Big).\leqno(9.14)$$
In other words, if $A_\lambda$, $B_\mu$ are the subschemes of
$\Omega_\lambda$, $\Omega'_\mu$ defined by the equations
$g_{\lambda,j}(x)=0$, $g'_{\mu,k}(y)=0$, where
$(g_{\lambda,j})$ and $(g'_{\mu,k})$ are generators of $\cJ_\lambda$
and $\cJ'_\mu$ respectively, then
$A_\lambda\times B_\mu$ is equipped with the structure sheaf
$\cO_{\Omega_\lambda\times\Omega'_\mu}\big/\big(
g_{\lambda,j}(x),g'_{\mu,k}(y)\big)$.

Now, let $\cS$ be a coherent sheaf over $\cO_X$ and let $\cS'$
be a coherent sheaf over $\cO_Y$. The (analytic) {\it external
tensor product} $\cS\stimes\cS'$ is defined to be
$$\cS\stimes\cS'={\rm pr}_1^\star\cS\otimes_{\cO_{X\times Y}}
{\rm pr}_2^\star\cS'.\leqno(9.15)$$
If we go back to the definition of the inverse image, we see that
the stalks of $\cS\stimes\cS'$ are given by
$$(\cS\stimes\cS')_{(x,y)}=\cO_{X\times Y,(x,y)}\otimes_
{\cO_{X,x}\otimes\cO_{Y,y}}(\cS_x\otimes_\bC\cS'_y)~,\leqno(9.15')$$
in particular $(\cS\stimes\cS')_{(x,y)}$ {\it does not coincide} with
the sheaf theoretic tensor product $\cS_x\otimes\cS'_y$ which is merely
a module over $\cO_{X,x}\otimes\cO_{Y,y}$. If $\cS$ and $\cS'$ are given
by local presentations
$$\cO_{X\restriction U}^{\oplus p}\buildo A\over\longrightarrow\cO_{X\restriction U}^{\oplus q}\longrightarrow
\cS_{\restriction U}\longrightarrow 0,~~~~
\cO_{Y\restriction U'}^{p'}\buildo B\over\longrightarrow\cO_{Y\restriction U'}^{q'}
\longrightarrow\cS'_{\restriction U'}\longrightarrow 0,$$
then $\cS\stimes\cS'$ is the coherent sheaf given by
$$\cO_{X\times Y\restriction U\times U'}^{pq'\oplus qp'}
\buildo{(A(x)\otimes\Id,\Id\otimes B(y))}\over{\larex 84 } 
\cO_{X\times Y\restriction U\times U'}^{qq'}\longrightarrow
(\cS\stimes\cS')_{\restriction U\times U'}\longrightarrow 0.$$

\subsection{9.8}{Zariski Embedding Dimension}
If $x$ is a point
of an analytic scheme $(X,\cO_X)$, the {\it Zariski embedding dimension}
of the germ $(X,x)$ is the smallest integer $N$ such that $(X,x)$ can be
embedded in $\bC^N$, i.e.\ such that there exists a patch of $X$
near $x$ isomorphic to $(A,\cO_\Omega/\cJ)$ where $\Omega$ is an open
subset of $\bC^N$. This dimension is denoted
$$\hbox{\rm embdim}(X,x)=~\hbox{\rm smallest~such}~~N.\leqno(9.16)$$
Consider the maximal ideal $\gm_{X,x}\subset\cO_{X,x}$ of functions which 
vanish at point~$x$. If $(X,x)$ is embedded in $(\Omega,x)=(\bC^N,0)$,
then $\gm_{X,x}/\gm_{X,x}^2$ is generated by $z_1,\ldots,z_N$, 
so $d=\dim\gm_{X,x}/\gm_{X,x}^2\le N$. Let $s_1,\ldots,s_d$
be germs in $\gm_{\Omega,x}$ which yield a basis of 
$\gm_{X,x}/\gm_{X,x}^2\simeq\gm_{\Omega,x}/(\gm^2_{\Omega,x}+\cJ_x)$.
We can write
$$z_j=\sum_{1\le k\le d}c_{jk}s_k+u_j+f_j,~~~c_{jk}\in\bC,~~u_j\in
\gm^2_{\Omega,x},~~f_j\in\cJ_x,~~1\le j\le n.$$
Then we find $dz_j=\sum c_{jk}\,ds_k(x)+df_j(x)$, so that the rank of the 
system of diffe\-rentials $\big(df_j(x)\big)$ is at least $N-d$. Assume 
for example that $df_1(x),\ldots,df_{N-d}(x)$ are linearly independant$\,$. 
By the implicit function theorem, the equations
$f_1=\ldots=f_{N-d}=0$ define a germ of smooth subvariety
$(Z,x)\subset(\Omega,x)$ of dimension $d$ which contains $(X,x)$.
We have $\cO_Z=\cO_\Omega/(f_1,\ldots,f_{N-d})$ in a neighborhood of $x$, thus
$$\cO_X=\cO_\Omega/\cJ\simeq\cO_Z/\cJ'~~~~\hbox{\rm where}~~
\cJ'=\cJ/(f_1,\ldots,f_{N-d}).$$
This shows that $(X,x)$ can be imbedded in $\bC^d$, and we get
$$\hbox{\rm embdim}(X,x)=\dim\gm_{X,x}/\gm_{X,x}^2.\leqno(9.17)$$

\claim{(9.18) Remark} \rm For a given dimension $n=\dim(X,x)$, the embedding
dimension $d$ can be arbitrarily large. Consider for example the curve
$\Gamma\subset\bC^N$ parametrized by $\bC\ni t\longmapsto(t^N,t^{N+1},\ldots,
t^{2N-1})$. Then $\cO_{\Gamma,0}$ is the ring of convergent
series in $\bC\{t\}$ which have no terms $t,t^2,\ldots,t^{N-1}$, and
$\gm_{\Gamma,0}/\gm_{\Gamma,0}^2$ admits precisely $z_1=\smash{t^N}
,\ldots,z_N=\smash{t^{2N-1}}$ as a basis. Therefore $n=1$ but $d=N$ can be as
large as we want.
\endclaim

\section{10}{Bimeromorphic maps, Modifications and Blow-ups}
It is a very frequent situation in analytic or algebraic geometry that
two complex spaces have isomorphic dense open subsets but are nevertheless
different along some analytic subset. These ideas are made precise by the
notions of modification and bimeromorphic map. This will also lead us to
generalize the notion of meromorphic function to maps between analytic
schemes. If $(X,\cO_X)$ is an analytic scheme, $\cM_X$ denotes the sheaf
of meromorphic functions on~$X$, defined at the beginning of \S~6.2.

\claim{(10.1) Definition} Let $(X,\cO_X)$, $(Y,\cO_Y)$ be analytic
schemes. An analytic morphism $F:X\to Y$ is said to be a modification
if $F$ is proper and if there exists a nowhere dense closed analytic
subset $B\subset Y$ such that the restriction $F:X\ssm F^{-1}(B)\to
Y\ssm B$ is an isomorphism.
\endclaim

\claim{(10.2) Definition} If $F:X\to Y$ is a modification, then the
comorphism $F^\star:f^\star\cO_Y\to\cO_X$ induces an isomorphism
$F^\star:f^\star\cM_Y\to\cM_X$ for the sheaves of meromorphic functions
on $X$ and~$Y$.
\endclaim

\proof{Proof} Let $v=g/h$ be a section of $\cM_Y$ on a small open set $\Omega$
where $u$ is actually given as a quotient of functions $g,h\in\cO_Y(\Omega)$.
Then $F^\star u=(g\circ F)/(h\circ F)$ is a section of $\cM_X$ on
$F^{-1}(\Omega)$, for $h\circ F$ cannot vanish identically on any open subset
$W$ of~$F^{-1}(\Omega)$ (otherwise $h$ would vanish on the open subset
$F(W\ssm F^{-1}(B))$ of $\Omega\ssm B$). Thus the extension of the comorphism
to sheaves of meromorphic functions is well defined. Our claim is that
this is an isomorphism. The injectivity of $F^\star$ is clear:
$F^\star u=0$ implies $g\circ F=0$, which implies $g=0$ on $\Omega\ssm B$
and thus $g=0$ on $\Omega$ because $B$ is nowhere dense. In order to prove
surjectivity, we need only show that every section $u\in\cO_X(F^{-1}(\Omega))$
is in the image of $\cM_Y(\Omega)$ by~$F^\star$. For this, we may shrink
$\Omega$ into a relatively compact subset $\Omega'\compact\Omega$ and
thus assume that $u$ is bounded (here we use the properness of $F$ through
the fact that $F^{-1}(\Omega')$ is relatively compact in $F^{-1}(\Omega))$.
Then $v=u\circ F^{-1}$ defines a bounded holomorphic function on~$\Omega\ssm
B$. By Th.~7.2, it follows that $v$ is weakly holomorphic for the reduced
structure of~$Y$. Our claim now follows from the following Lemma.\qed
\endproof

\claim{(10.3) Lemma} If $(X,\cO_X)$ is an analytic scheme, then every
holomorphic function $v$ in the complement of a nowhere dense analytic subset
$B\subset Y$ which is weakly holomorphic on~$X_\red$ is meromorphic
on~$X$.
\endclaim

\proof{Proof} It is enough to argue with the germ of $v$ at any point
$x\in Y$, and thus we may suppose that $(Y,\cO_Y)=(A,\cO_\Omega/\cI)$
is embedded in~$\bC^N$. Because $v$ is weakly holomorphic, we can
write $v=g/h$ in $Y_\red$, for some germs of holomorphic functions
$g,h$. Let $\wt g$ and $\wt h$ be extensions of $g$, $h$ to
$\cO_{\Omega,x}$. Then there is a neighborhood $U$ of $x$ such that
$\wt g-v\wt h$ is a nilpotent section of $cO_\Omega(U\ssm B)$ which is in $\cI$ on
\endproof

\claim{(10.4) Definition}
A meromorphic map $F:X\merto Y$ is a scheme morphism $F:X\ssm A\to Y$ defined
in the complement of a nowhere dense analytic subset $A\subset X$, such that
the closure of the graph of $F$ in $X\times Y$ is an analytic subset $($for
the reduced complex space structure of $X\times Y)$.
\endclaim

\section{11}{Exercises}\begineightpoint

\subsubsection{11.1}{} Let $\cA$ be a sheaf on a topological space~$X$. If the
sheaf space $\wt\cA$ is Hausdorff, show that $\cA$ satisfies the
following {\it unique continuation principle}: any two sections
$s,s'\in\cA(U)$ on a connected open set $U$ which coincide on some non empty
open subset~$V\subset U$ must coincide identically on~$U$. Show that the
converse holds if $X$ is Hausdorff and locally connected.

\subsubsection{11.2}{} Let $\cA$ be a sheaf of abelian groups on~$X$ and let
$s\in\cA(X)$. The support of~$s$, denoted $\Supp\,s$, is defined to be
$\{x\in X\,;\,s(x)\ne 0\}$. Show that $\Supp\,s$ is a closed subset
of~$X$. The support of $\cA$ is defined to be $\Supp\,\cA=\{x\in
X\,;\,\cA_x\ne 0\}$. Show that $\Supp\,\cA$ is not necessarily closed:
if $\Omega$ is an open set in $X$, consider the sheaf $\cA$ such that
$\cA(U)$ is the set of continuous functions $f\in\cC(U)$ which vanish
on a neighborhood of $U\cap(X\ssm\Omega)$.

\subsubsection{11.3}{} Let $\cA$ be a sheaf of rings on a topological space~$X$
and let $\cF$, $\cG$ be sheaves of $\cA$-modules. We define a presheaf
$\cH=\cHom_\cA(\cF,\cG)$ such that $\cH(U)$ is the module of all
sheaf-homomorphisms $\cF_{\restriction U}\to\cG_{\restriction U}$ which are
$\cA$-linear.\smallskip
\item{a)} Show that $\cHom_\cA(\cF,\cG)$ is a sheaf and that there exists a
canonical homomorphism\break $\varphi_x:\cHom_\cA(\cF,\cG)_x\longrightarrow
\hom_{\cA_x}(\cF_x,\cG_x)$ for every $x\in X$.
\smallskip
\item{b)} If $\cF$ is locally finitely generated, then $\varphi_x$ is
injective, and if $\cF$ has local finite presentations as in (3.12), then
$\varphi_x$ is bijective.
\smallskip
\item{c)} Suppose that $\cA$ is a coherent sheaf of rings and that
$\cF$, $\cG$ are coherent modules over~$\cA$. Then $\cHom_\cA(\cF,\cG)$ is
a coherent $\cA$-module.\\
{\it Hint}\/: observe that the result is true if $\cF=\cA^{\oplus p}$ and use a local
presentation of $\cF$ to get the conclusion.

\subsubsection{11.4}{} Let $f:X\to Y$ be a continuous map of topological spaces.
Given sheaves of abelian groups $\cA$ on $X$ and $\cB$ on~$Y$, show that
there is a natural isomorphism
$$\hom_X(f^{-1}\cB,\cA)=\hom_Y(\cB,f_\star\cA).$$
{\it Hint}\/: use the natural morphisms (2.17).

\subsubsection{11.5}{} Show that the sheaf of polynomials over $\bC^n$ is
a coherent sheaf of rings (with either the ordinary topology or the
Zariski topology on $\bC^n$). Extend this result to the case of
regular algebraic functions on an algebraic variety.\\
{\it Hint}\/: check that the proof of the Oka theorem still applies.

\subsubsection{11.6}{} Let $P$ be a non zero polynomial on~$\bC^n$.
If $P$ is irreducible in $\bC[z_1,\ldots,z_n]$, show that the
hypersurface $H=\smash{P^{-1}(0)}$ is globally irreducible as an analytic set.
In general, show that the irreducible components of $H$ are in a
one-to-one correspondence with the irreducible factors of~$P$.\\
{\it Hint}\/: for the first part, take coordinates such that
$P(0,\ldots,0,z_n)$ has degree equal to $P\,$; if $H$ splits in two
components $H_1$, $H_2$, then $P$ can be written as a product
$P_1P_2$ where the roots of $P_j(z',z_n)$ correspond to points in~$H_j$.

\subsubsection{11.7}{} Prove the following facts:\smallskip
\item{a)} For every algebraic variety $A$ of pure dimension $p$ in~$\bC^n$,
there are coordinates $z'=(z_1,\ldots,z_p)$, $z''=(z_{p+1},\ldots,z_n)$ such that
$\pi:A\to\smash{\bC^p}$, $z\mapsto z''$ is proper with finite fibers, and
such that $A$ is entirely contained in a cone
$$|z''|\le C(|z'|+1).$$
{\it Hint}\/: imitate the proof of Cor.~4.11.
\smallskip
\item{b)} Conversely if an analytic set $A$ of pure dimension $p$ in $\bC^n$
is contained in a cone $|z''|\le C(|z'|+1)$, then $A$ is algebraic.\\
{\it Hint}\/: first apply $(5.9)$ to conclude that the projection $\pi:
A\to\bC^p$ is finite. Then repeat the arguments used in the final part
of the proof of Th.~4.19.
\item{c)} Deduce from a), b) that an algebraic set in $\bC^n$ is
irreducible if and only if it is irreducible as an analytic set.

\subsubsection{11.8}{} Let $\Gamma:f(x,y)=0$ be a germ of analytic curve in
$\bC^2$ through~$(0,0)$ and let $(\Gamma_j,0)$ be the irreducible
components of $(\Gamma,0)$. Suppose that $f(0,y)\not\equiv 0$. Show that
the roots $y$ of $f(x,y)=0$ corresponding to points of $\Gamma$ near $0$
are given by {\it Puiseux expansions} of the form $y=g_j(x^{1/q_j})$,
where $g_j\in\cO_{\bC,0}$ and where $q_j$ is the sheet number of the
projection $\Gamma_j\to\bC$, $(x,y)\mapsto x$.\\
{\it Hint}\/: special case of the parametrization obtained in~(4.27).

\subsubsection{11.9}{} The goal of this exercise is to prove the existence and
the analyticity of the {\it tangent cone} to an arbitrary analytic
germ $(A,0)$ in~$\bC^n$. Suppose
that $A$ is defined by holomorphic equations $f_1=\ldots=f_N=0$ in
a ball $\Omega=B(0,r)$. Then the (set theoretic) tangent cone to $A$
at $0$ is the set $C(A,0)$ of all limits of sequences $t_\nu^{-1}z_\nu$
with $z_\nu\in A$ and $\bC^\star\ni t_\nu$ converging to~$0$.\smallskip
\item{a)} Let $E$ be the set of points $(z,t)\in\Omega\times\bC^\star$
such that $z\in t^{-1}A$. Show that the closure $\smash{\ol E}$ in
$\Omega\times\bC$ is analytic.\\
{\it Hint}\/: observe that $E=A\ssm(\Omega\times\{0\})$ where $A=\{f_j(tz)=0\}$
and apply Cor.~5.4.\smallskip
\item{b)} Show that $C(A,0)$ is a conic set and that $\ol E\cap(\Omega\times
\{0\})=C(A,0)\times\{0\}$ and conclude. Infer from this that $C(A,0)$ is an
algebraic subset of~$\bC^n$.

\subsubsection{11.10}{} Give a new proof of Theorem~5.5 based on the coherence
of ideal sheaves and on the strong noetherian property.

\subsubsection{11.11}{} Let $X$ be an analytic space and let $A$, $B$ be
analytic subsets of pure dimensions. Show that
$\codim_X(A\cap B)\le\codim_X A+\codim_X B$
if $A$ or $B$ is a local complete intersection, but that the equality
does not necessarily hold in general.\\
{\it Hint}\/: see Remark~(6.5).

\subsubsection{11.12}{} Let $\Gamma$ be the curve in $\bC^3$ parametrized by
$\bC\ni t\longmapsto(x,y,z)=(t^3,t^4,t^5)$.
Show that the ideal sheaf $\cI_\Gamma$ is generated by the polynomials
$xz-y^2$, $x^3-yz$ and $x^2y-z^2$, and that the germ $(\Gamma,0)$ is
not a (sheaf theoretic) complete intersection.\\
{\it Hint}\/: $\Gamma$ is smooth except at the origin. Let $f(x,y,z)=
\sum a_{\alpha\beta\gamma}x^\alpha y^\beta z^\gamma$ be a convergent
power series near~$0$. Show that $f\in\cI_{\Gamma,0}$ if and only if
all weighted homogeneous components 
$$f_k=\sum_{3\alpha+4\beta+5\gamma=k}
a_{\alpha\beta\gamma}x^\alpha y^\beta z^\gamma$$ are in~$\cI_{\Gamma,0}$.
By means of suitable substitutions, reduce the proof to the case when
$f=f_k$ is homogeneous with all non zero monomials satisfying
$\alpha\le 2$, $\beta\le 1$, $\gamma\le 1$; then check that there is
at most one such monomial in each weighted degree${}\le 15$ the product
of a power of $x$ by a homogeneous polynomial of weighted degree~$\le 8$.
\endeightpoint


\chapter{Chapter III}{Positive Currents and Lelong Numbers}

\begineightpoint
In 1957, P. Lelong introduced natural positivity concepts for currents
of pure bidimension $(p,p)$ on complex manifolds. With every analytic
subset is associated a current of integration over its set of regular
points and all such currents are positive and closed. The important
closedness property is proved here via the Skoda-El Mir extension
theorem. Positive currents have become an important tool for the study
of global geometric problems as well as for questions related to local
algebra and intersection theory. We develope here a differential
geometric approach to intersection theory through a detailed study of
wedge products of closed positive currents (Monge-Amp\`ere operators).
The Lelong-Poincar\'e equation and the Jensen-Lelong formula are basic
in this context, providing a useful tool for studying the location and
multiplicities of zeroes of entire functions on $\bC^n$ or on a
manifold, in relation with the growth at infinity. Lelong numbers of
closed positive currents are then introduced; these numbers can be seen
as a generalization to currents of the notion of multiplicity of a germ
of analytic set at a singular point. We prove various properties of
Lelong numbers (e.g.\ comparison theorems, semi-continuity theorem of
Siu, transformation under holomorphic maps). As an application to
Number Theory, we prove a general Schwarz lemma in $\bC^n$ and derive
from it Bombieri's theorem on algebraic values of meromorphic maps and
the famous theorems of Gelfond-Schneider and Baker on the transcendence
of exponentials and logarithms of algebraic numbers.
\endeightpoint

\section{1}{Basic Concepts of Positivity}
\subsection{1.A}{Positive and Strongly Positive Forms}
Let $V$ be a complex vector space of dimension $n$ and $(z_1,\ldots,z_n)$
coordinates on $V$. We denote by $(\partial/\partial z_1,\ldots,
\partial/\partial z_n)$ the corresponding basis of $V$,
by $(dz_1,\ldots,dz_n)$ its dual basis in $V^\star$ and consider
the exterior algebra
$$\Lambda V^\star_\bC=\bigoplus\Lambda^{p,q}V^\star,~~~~
\Lambda^{p,q}V^\star=\Lambda^pV^\star\otimes\Lambda^q\ol{V^\star}.$$
We are of course especially interested in the case where $V=T_xX$
is the tangent space to a complex manifold $X$, but we want to
emphasize here that our considerations only involve linear algebra.
Let us first observe that $V$ has a canonical orientation,
given by the $(n,n)$-form
$$\tau(z)=\ii dz_1\wedge d\ol z_1\wedge\ldots\wedge\ii dz_n\wedge d\ol z_n=
2^n\,dx_1\wedge dy_1\wedge\ldots\wedge dx_n\wedge dy_n$$
where $z_j=x_j+iy_j$. In fact, if $(w_1,\ldots,w_n)$ are other coordinates,
we find
$$\eqalign{
&dw_1\wedge\ldots\wedge dw_n=\det(\partial w_j/\partial z_k)\,dz_1\wedge\ldots
\wedge dz_n,\cr
&\tau(w)=\big|\det(\partial w_j/\partial z_k)\big|^2\,\tau(z).\cr}$$
In particular, a complex manifold always has a canonical orientation.
More generally, natural positivity concepts for $(p,p)$-forms can be
defined.

\claim{(1.1) Definition} A $(p,p)$-form $u\in\Lambda^{p,p}V^\star$
is said to be positive if for all $\alpha_j\in V^\star$,
$1\le j\le q=n-p$, then
$$u\wedge \ii\alpha_1\wedge\ol\alpha_1\wedge\ldots\wedge \ii\alpha_q\wedge
\ol\alpha_q$$
is a positive $(n,n)$-form. A $(q,q)$-form $v\in\Lambda^{q,q}V^\star$
is said to be strongly positive if $v$ is a convex combination
$$v=\sum\gamma_s~\ii\alpha_{s,1}\wedge\ol\alpha_{s,1}
\wedge\ldots\wedge \ii\alpha_{s,q}\wedge\ol\alpha_{s,q}$$
where $\alpha_{s,j}\in V^\star$ and $\gamma_s\ge 0$.
\endclaim

\claim{(1.2) Example} \rm Since $\ii^p(-1)^{p(p-1)/2}=\ii^{p^2}$, we have the 
commutation rules
$$\eqalign{
&\ii\alpha_1\wedge\ol\alpha_1\wedge\ldots\wedge \ii\alpha_p\wedge\ol\alpha_p=
i^{p^2}\alpha\wedge\ol\alpha,~~~~\forall\alpha=\alpha_1\wedge\ldots\wedge\alpha_p
\in\Lambda^{p,0}V^\star,\cr
&i^{p^2}\beta\wedge\ol \beta\wedge i^{m^2}\gamma\wedge\ol \gamma=
i^{(p+m)^2}\beta\wedge \gamma\wedge\ol{\beta\wedge \gamma},~~~~
\forall \beta\in\Lambda^{p,0}V^\star,~\forall \gamma\in\Lambda^{m,0}V^\star.\cr}$$
Take $m=q$ to be the complementary degree of $p$. Then
$\beta\wedge \gamma=\lambda dz_1\wedge\ldots\wedge dz_n$ for some
$\lambda\in\bC$ and $i^{n^2}\beta\wedge\gamma\wedge\ol{\beta\wedge \gamma}=
|\lambda|^2\tau(z)$. If we set $\gamma=\alpha_1\wedge\ldots\wedge\alpha_q$, 
we find that $i^{p^2}\beta\wedge\ol \beta$ is a positive $(p,p)$-form 
for every $\beta\in\Lambda^{p,0}V^\star\,$; in
particular, strongly positive forms are positive.\qed
\endclaim

The sets of positive and strongly positive forms are closed convex cones,
i.e.\ closed and stable under convex combinations. By definition, the positive
cone is dual to the strongly positive cone via the pairing
$$\cmalign{
&\Lambda^{p,p}V^\star\times&\Lambda^{q,q}V^\star&\longrightarrow\bC\cr
&\hfill(u,&v)&\longmapsto u\wedge v/\tau,\cr}\leqno(1.3)$$
that is, $u\in\Lambda^{p,p}V^\star$ is positive if and only if 
$u\wedge v\ge 0$ for all strongly positive forms
$v\in\Lambda^{q,q}V^\star$. Since the bidual of an arbitrary convex cone
$\Gamma$ is equal to its closure $\ol\Gamma$, we also obtain that
$v$ is strongly positive if and only if $v\wedge u=u\wedge v$ is $\ge 0$
for all positive forms $u$. Later on, we will need the following
elementary lemma.

\claim{(1.4) Lemma} Let $(z_1,\ldots,z_n)$ be arbitrary coordinates on $V$.
Then $\Lambda^{p,p}V^\star$ admits a basis consisting of strongly 
positive forms
$$\beta_s=\ii\beta_{s,1}\wedge\ol\beta_{s,1}\wedge\ldots
\wedge\ii\beta_{s,p}\wedge\ol\beta_{s,p},~~~~1\le s\le{n\choose p}^2$$
where each $\beta_{s,l}$ is of the type $dz_j\pm dz_k$ or
$dz_j\pm \ii dz_k$, $1\le j,k\le n$.
\endclaim

\proof{Proof} Since one can always extract a basis from a set of generators, 
it is sufficient to see that the family of forms of the above
type generates $\Lambda^{p,p}V^\star$. This follows from the identities
$$\cmalign{
&4dz_j\wedge d\ol z_k=~~~&(dz_j+dz_k)&\wedge&(\ol{dz_j+dz_k})&-
&(dz_j-dz_k)&\wedge&(\ol{dz_j-dz_k})\cr
&\hfill+\ii&(dz_j+\ii dz_k)&\wedge&(\ol{dz_j+\ii dz_k})&-\ii&(dz_j-\ii
dz_k)&\wedge&(\ol{dz_j-\ii dz_k}),\cr}$$
$$dz_{j_1}\wedge\ldots\wedge dz_{j_p}\wedge d\ol z_{k_1}\wedge
\ldots\wedge d\ol z_{k_p}=\pm\bigwedge_{1\le s\le p}dz_{j_s}\wedge d\ol
z_{k_s}.\eqno{\square}$$
\endproof

\claim{(1.5) Corollary} All positive forms $u$ are real, i.e.\ satisfy $\ol u=u$.
In terms of coordinates, if $u=i^{p^2}\sum_{|I|=|J|=p}u_{I,J}\,
dz_I\wedge d\ol z_J$, then the coefficients satisfy the hermitian 
symmetry relation $\ol{u_{I,J}}=u_{J,\,I}$.
\endclaim

\proof{Proof} Clearly, every strongly positive $(q,q)$-form is real. By Lemma
1.4, these forms generate over $\bR$ the real elements of 
$\Lambda^{q,q}V^\star$, so we conclude by duality that positive
$(p,p)$-forms are also real.\qed
\endproof

\claim{(1.6) Criterion} A form $u\in\Lambda^{p,p}V^\star$ is 
positive if and only if its restriction $u_{\restriction S}$ to every 
$p$-dimensional subspace $S\subset V$ is a positive volume form on $S$.
\endclaim

\proof{Proof} If $S$ is an arbitrary $p$-dimensional subspace of $V$
we can find linear coordinates $(z_1,\ldots,z_n)$ on $V$ such that
$S=\{z_{p+1}=\ldots=z_n=0\}$. Then 
$$u_{\restriction S}=\lambda_S\,\ii dz_1\wedge d\ol z_1\wedge\ldots\wedge
\ii dz_p\wedge d\ol z_p$$
where $\lambda_S$ is given by
$$u\wedge\ii dz_{p+1}\wedge d\ol z_{p+1}\wedge\ldots\wedge
\ii dz_n\wedge d\ol z_n=\lambda_S\,\tau(z).$$
If $u$ is positive then $\lambda_S\ge 0$ so $u_{\restriction S}$ is
positive for every $S$. The converse is true because the $(n-p,n-p)$-forms
$\bigwedge_{j>p}\ii dz_j\wedge d\ol z_j$ generate all strongly positive
forms when $S$ runs over all $p$-dimensional subspaces.\qed
\endproof

\claim{(1.7) Corollary} A form $u=\ii\sum_{j,k}u_{jk}\,dz_j\wedge 
d\ol z_k$ of bidegree $(1,1)$ is positive if and only if 
$\xi\mapsto\sum u_{jk}\xi_j\ol\xi_k$ is a semi-positive hermitian form
on $\bC^n$.
\endclaim

\proof{Proof} If $S$ is the complex line generated by $\xi$ and $t\mapsto t\xi$
the parametrization of $S$, then $u_{\restriction S}=
\big(\sum u_{jk}\xi_j\ol\xi_k\big)\,\ii dt\wedge d\ol t$.\qed
\endproof

Observe that there is a canonical one-to-one correspondence between hermitian 
forms and real $(1,1)$-forms on $V$. The correspondence is given by
$$h=\sum_{1\le j,k\le n}h_{jk}(z)\,dz_j\otimes d\ol z_k\longmapsto
u=\ii\sum_{1\le j,k\le n}h_{jk}(z)\,dz_j\wedge d\ol z_k\leqno(1.8)$$
and does not depend on the choice of coordinates: indeed, as $\ol h_{jk}=
h_{kj}$, one finds immediately
$$u(\xi,\eta)=\ii\sum h_{jk}(z)(\xi_j\ol\eta_k-\eta_j\ol\xi_k)=
-2\Im h(\xi,\eta),~~~~\forall\xi,\eta\in TX.$$
Moreover, $h$ is $\ge 0$ as a hermitian form if and only if $u\ge 0$
as a $(1,1)$-form. A diagonalization of $h$ shows that every positive 
$(1,1)$-form 
$u\in\Lambda^{1,1}V^\star$ can be written
$$u=\sum_{1\le j\le r}\ii\gamma_j\wedge\ol\gamma_j~,~~~~\gamma\in V^\star,
~~r={\rm rank~of~~}u,$$
in particular, every positive $(1,1)$-form is strongly positive. By 
duality, this is also true for $(n-1,n-1)$-forms.

\claim{(1.9) Corollary} The notions of positive and strongly positive
$(p,p)$-forms coincide for $p=0,1,n-1,n$.\qed
\endclaim

\claim{(1.10) Remark} \rm It is not difficult to see, however, that positivity and
strong positivity differ in all bidegrees $(p,p)$ such that
$2\le p\le n-2$. Indeed, a positive form $i^{p^2}\beta\wedge\ol\beta$
with $\beta\in\Lambda^{p,0}V^\star$ is strongly positive if and only if
$\beta$ is decomposable as a product $\beta_1\wedge\ldots\wedge\beta_p$.
To see this, suppose that 
$$i^{p^2}\beta\wedge\ol\beta=\sum_{1\le j\le N}i^{p^2}\gamma_j\wedge\ol\gamma_j$$
where all $\gamma_j\in\Lambda^{p,0}V^\star$ are decomposable. Take $N$
minimal. The equality can be also written as an equality of hermitian
forms $|\beta|^2=\sum|\gamma_j|^2$ if $\beta,\gamma_j$ are seen as
linear forms on $\Lambda^pV$. The hermitian form $|\beta|^2$ has rank one,
so we must have $N=1$ and $\beta=
\lambda\gamma_j$, as desired. Note that there are many non decomposable 
$p$-forms in all degrees $p$ such that $2\le p\le n-2$, e.g.\
$(dz_1\wedge dz_2+dz_3\wedge dz_4)\wedge dz_5\wedge\ldots\wedge dz_{p+2}\,$:
if a $p$-form is decomposable, the vector space of its contractions
by elements of $\bigwedge^{p-1}V$ is a $p$-dimensional subspace
of $V^\star$; in the above example the dimension is $p+2$.
\endclaim

\claim{(1.11) Proposition} If $u_1,\ldots,u_s$ are positive forms, all
of them strongly positive $($resp.\ all except perhaps one$)$, then
$u_1\wedge\ldots\wedge u_s$ is strongly positive $($resp.\ positive$)$.
\endclaim

\proof{Proof} Immediate consequence of Def.~1.1. Observe however that
the wedge product of two positive forms is not positive in general
(otherwise we would infer that positivity coincides with strong
positivity).\qed
\endproof

\claim{(1.12) Proposition} If $\Phi:W\longrightarrow V$ is a complex linear
map and $u\in\Lambda^{p,p}V^\star$ is $($strongly$)$ positive,
then $\Phi^\star u\in\Lambda^{p,p}W^\star$ is $($strongly$)$ positive.
\endclaim

\proof{Proof} This is clear for strong positivity, since
$$\Phi^\star(\ii\alpha_1\wedge\ol\alpha_1\wedge\ldots\wedge
\ii\alpha_p\wedge\ol\alpha_p)=\ii\beta_1\wedge\ol\beta_1\wedge\ldots\wedge
\ii\beta_p\wedge \ol\beta_p$$
with $\beta_j=\Phi^\star\alpha_j\in W^\star$, for all $\alpha_j\in V^\star$. 
For $u$ positive, we may apply Criterion~1.6: if $S$ is a $p$-dimensional
subspace of $W$, then $u_{\restriction\Phi(S)}$ and 
$(\Phi^\star u)_{\restriction S}=(\Phi_\restriction S)^\star 
u_{\restriction\Phi(S)}$ are positive when $\Phi_{\restriction S}:
S\longrightarrow\Phi(S)$ is an isomorphism; otherwise we get 
$(\Phi^\star u)_{\restriction S}=0$.\qed
\endproof

\subsection{1.B}{Positive Currents} 
The duality between the positive and strongly positive cones of forms
can be used to define corresponding positivity notions for currents. 

\claim{(1.13) Definition} A current $T\in\cD'_{p,p}(X)$ is said to be positive
$($resp.\ strongly positive$)$ if $\langle T,u\rangle\ge 0$ for all
test forms $u\in\cD_{p,p}(X)$ that are strongly positive $($resp.\ positive$)$
at each point. The set of positive $($resp.\ strongly positive$)$ currents
of bidimension $(p,p)$ will be denoted
$$\cD^{\prime+}_{p,p}(X),~~~~\hbox{\rm resp.}~~\cD^{\prime\oplus}_{p,p}(X).$$
\endclaim

It is clear that (strong) positivity is a local property and that the sets
$\cD^{\prime\oplus}_{p,p}(X)\subset\cD^{\prime+}_{p,p}(X)$ are closed convex cones
with respect to the weak topology. Another way of stating Def.~1.13 is:
\medskip
\noindent{\it $T$ is positive $($strongly positive$)$ if and only if 
$T\wedge u\in\cD'_{0,0}(X)$ is a positive
measure for all strongly positive $($positive$)$ forms 
$u\in \cC^\infty_{p,p}(X)$.}
\medskip
\noindent This is so because a distribution $S\in\cD'(X)$ such that $S(f)\ge 0$
for every non-negative function $f\in\cD(X)$ is a positive measure. 

\claim{(1.14) Proposition} Every positive current 
$T=i^{(n-p)^2}\sum T_{I,J}\,dz_I
\wedge d\ol z_J$ in $\cD^{\prime +}_{p,p}(X)$ is real and of order $0$,
i.e.\ its coefficients $T_{I,J}$ are complex measures and satisfy
$\ol{T_{I,J}}=T_{J,\,I}$ for all multi-indices $|I|=|J|=n-p$. Moreover 
$T_{I,I}\ge 0$, and the absolute values $|T_{I,J}|$ of
the measures $T_{I,J}$ satisfy the inequality
$$\lambda_I\lambda_J\,|T_{I,J}|\le 2^p\sum_M~\lambda_M^2\,T_{M,M},~~~~
I\cap J\subset M\subset I\cup J$$
where $\lambda_k\ge 0$ are arbitrary coefficients and 
$\lambda_I=\prod_{k\in I}\lambda_k$.
\endclaim

\proof{Proof} Since positive forms are real, positive currents have to be real
by duality. Let us denote by $K=\complement I$ and $L=\complement J$ the
ordered complementary multi-indices of $I,J$ in $\{1,2,\ldots,n\}$. The 
distribution $T_{I,I}$ is a positive measure because
$$T_{I,I}~\tau=T\wedge i^{p^2}dz_K\wedge d\ol z_K\ge 0.$$
On the other hand, the proof of Lemma~1.4 yields
$$\eqalign{
T_{I,J}~\tau&=\pm\,T\wedge i^{p^2}dz_K\wedge d\ol z_L=
\sum_{a\in(\bZ/4\bZ)^p}\varepsilon_a\,T\wedge\gamma_a
~~~~{\rm where}\cr
\gamma_a&=\bigwedge_{1\le s\le p}{\ii\over 4}(dz_{k_s}+
i^{a_s}dz_{l_s})\wedge(\ol{dz_{k_s}+i^{a_s}dz_{l_s}}),~~~~
\varepsilon_a=\pm 1,\pm i.\cr}$$
Now, each $T\wedge\gamma_a$ is a positive measure, hence $T_{I,J}$ is a 
complex measure and
$$\eqalign{
|T_{I,J}|\,\tau&\le\sum_a T\wedge\gamma_a=
T\wedge\sum_a\gamma_a\cr
&=T\wedge\bigwedge_{1\le s\le p}\Big(\sum_{a_s\in\bZ/4\bZ}{\ii\over 4}
(dz_{k_s}+i^{a_s}dz_{l_s})\wedge(\ol{dz_{k_s}+i^{a_s}dz_{l_s}})
\Big)\cr
&=T\wedge\bigwedge_{1\le s\le p}\big(\ii dz_{k_s}\wedge d\ol z_{k_s}+
\ii dz_{l_s}\wedge d\ol z_{l_s}\big).\cr}$$
The last wedge product is a sum of at most $2^p$ terms, each of which is of 
the type $i^{p^2}dz_M\wedge d\ol z_M$ with $|M|=p$ and $M\subset K\cup L$.
Since $T\wedge i^{p^2}dz_M\wedge d\ol z_M=T_{\complement M,\complement M}\,\tau$
and $\complement M\supset\complement K\cap\complement L=I\cap J$, we find
$$|T_{I,J}|\le 2^p\sum_{M\supset I\cap J}T_{M,M}.$$
Now, consider a change of coordinates $(z_1,\ldots,z_n)=\Lambda w=
(\lambda_1 w_1,\ldots,\lambda_nw_n)$ with $\lambda_1,\ldots,\lambda_n>0$.
In the new coordinates, the current $T$ becomes $\Lambda^\star T$
and its coefficients become $\lambda_I\lambda_J\,T_{I,J}(\Lambda w)$.
Hence, the above inequality implies
$$\lambda_I\lambda_J\,|T_{I,J}|\le 2^p\sum_{M\supset I\cap J}
\lambda_M^2\,T_{M,M}.$$
This inequality is still true for $\lambda_k\ge 0$ by passing to the limit.
The inequality of Prop.~1.14 follows when all coefficients $\lambda_k$,
$k\notin I\cup J$, are replaced by $0$, so that $\lambda_M=0$ for 
$M\not\subset I\cup J$.\qed
\endproof

\claim{(1.15) Remark} \rm If $T$ is of order $0$, we define the {\it mass
measure} of $T$ by $\|T\|=\sum|T_{I,J}|$ (of course $\|T\|$ depends on 
the choice of coordinates). By the  Radon-Nikodym theorem, we can write 
$T_{I,J}=f_{I,J}\|T\|$ with a Borel function $f_{I,J}$ such that 
$\sum|f_{I,J}|=1$. Hence
$$T=\|T\|\,f,~~~~{\rm where}~~f=i^{(n-p)^2}\sum f_{I,J}\,dz_I\wedge d\ol z_J.$$
Then $T$ is (strongly) positive if and only if the form $f(x)\in
\Lambda^{n-p,n-p}T^\star_xX$ is (strongly) positive at $\|T\|$-almost 
all points $x\in X$.
Indeed, this condition is clearly sufficient. On the other hand, if 
$T$ is (strongly) positive and $u_j$ is a sequence of forms with
constant coefficients in $\Lambda^{p,p}T^\star X$ which is dense in 
the set of strongly positive (positive) forms, then
$T\wedge u_j=||T||\,f\wedge u_j$, so $f(x)\wedge u_j$ has to be a positive
$(n,n)$-form except perhaps for $x$ in a set $N(u_j)$ of $\|T\|$-measure~$0$.
By a simple density argument, we see that $f(x)$ is (strongly) positive 
outside the $\|T\|$-negligible set $N=\bigcup N(u_j)$.

As a consequence of this proof, $T$ is positive (strongly positive) if 
and only if $T\wedge u$ is a positive measure for all strongly positive 
(positive) forms $u$ of bidegree 
$(p,p)$ with {\it constant coefficients} in the given coordinates 
$(z_1,\ldots,z_n)$. It follows that if $T$ is (strongly) positive
in a coordinate patch $\Omega$, then the convolution
$T\star\rho_\varepsilon$ is (strongly) positive in $\Omega_\varepsilon=
\{x\in\Omega\,;\,d(x,\partial\Omega)>\varepsilon\}$.\qed
\endclaim

\claim{(1.16) Corollary} If $T\in\cD'_{p,p}(X)$ and $v\in C^0_{s,s}(X)$ 
are positive, one of them $($resp.\ both of them$)$ strongly positive, 
then the wedge product $T\wedge v$ is a positive $($resp.\ strongly
positive$)$ current.
\endclaim

This follows immediately from Remark~1.15 and Prop.~1.11 for forms.
Similarly, Prop.~1.12 on pull-backs of positive forms easily shows 
that positivity properties of currents are 
preserved under direct or inverse images by holomorphic maps.

\claim{(1.17) Proposition} Let $\Phi:X\longrightarrow Y$ be a holomorphic map
between complex analytic manifolds.
\medskip
\item{\rm a)} If $T\in\cD^{\prime+}_{p,p}(X)$ and $\Phi_{\restriction
{\rm Supp}\,T}$ is proper, then $\Phi_\star T\in\cD^{\prime+}_{p,p}(Y)$.
\medskip
\item{\rm b)} If $T\in\cD^{\prime+}_{p,p}(Y)$ and if $\Phi$ is
a submersion with $m$-dimensional fibers, then
$\,\Phi^\star T\in\cD^{\prime+}_{p+m,p+m}(X)$.
\medskip
\noindent Similar properties hold for strongly positive currents.\qed
\endclaim

\subsection{1.C}{Basic Examples of Positive Currents}

We present here two fundamental examples which will be of interest 
in many circumstances.

\subsubsection{1.18}{Current Associated to a Plurisubharmonic Function.}
Let $X$ be a complex manifold and 
$u\in\Psh(X)\cap L^1_\loc(X)$ a plurisubharmonic function. Then
$$T=\ii d'd''u=\ii\sum_{1\le j,k\le n}{\partial^2u\over\partial 
z_j\partial\ol z_k}\,dz_j\wedge d\ol z_k$$
is a positive current of bidegree $(1,1)$. Moreover $T$ is closed
(we always mean here $d$-closed, that is, $dT=0$).
Assume conversely that $\Theta$ is a closed real $(1,1)$-current on
$X$. Poincar\'e's lemma implies that every point $x_0\in X$ has a
neighborhood $\Omega_0$ such that $\Theta=dS$ with 
$S\in\cD'_1(\Omega_0,\bR)$. Write $S=S^{1,0}+S^{0,1}$, where $S^{0,1}=\ol{S^{1,0}}$. Then
$d''S=\Theta^{0,2}=0$, and the Dolbeault-Grothendieck lemma shows that
$S^{0,1}=d''v$ on some neighborhood $\Omega\subset\Omega_0$, with
$v\in\cD'(\Omega,\bC)$. Thus
$$\eqalign{
S&=\ol{d''v}+d''v=d'\ol v+d''v,\cr
\Theta&=dS=d'd''(v-\ol v)=\ii d'd''u,\cr}$$
where $u=2\Re v\in\cD'(\Omega,\bR)$. If $\Theta\in \cC^\infty_{1,1}(X)$,
the hypoellipticity of $d''$ in bidegree $(p,0)$ shows that $d'u$ is 
of class $\cC^\infty$, so $u\in \cC^\infty(\Omega)$. When $\Theta$ is positive, 
the distribution $u$ is a plurisubharmonic function (Th.~I.3.31). 
We have thus proved:

\claim{(1.19) Proposition} If $\Theta\in\cD^{\prime+}_{n-1,n-1}(X)$
is a closed positive current of bidegree $(1,1)$, then for every point 
$x_0\in X$ there exists a neighborhood $\Omega$ of $x_0$ and $u\in\Psh(\Omega)$
such that $\Theta=\ii d'd''u$.\qed
\endclaim

\subsubsection{1.20}{Current of Integration on a Complex Submanifold.}
Let $Z\subset X$ be a closed $p$-dimensional 
complex submanifold with its canonical orientation and $T=[Z]$. Then
$T\in\cD^{\prime\oplus}_{p,p}(X)$. Indeed, every $(r,s)$-form of
total degree $r+s=2p$ has zero restriction to $Z$ unless $(r,s)=(p,p)$,
therefore we have $[Z]\in\cD'_{p,p}(X)$. Now, if $u\in\cD_{p,p}(X)$ is a
positive test form, then $u_{\restriction Z}$ is a positive volume form
on $Z$ by Criterion~1.6, therefore
$$\langle[Z],u\rangle=\int_Z u_{\restriction Z}\ge 0.$$
In this example the current $[Z]$ is also closed, because
$d[Z]=\pm [\partial Z]=0$ by Stokes' theorem.\qed

\subsection{1.D}{Trace Measure and Wirtinger's Inequality}
We discuss now some questions related to the concept of area on complex 
submanifolds. Assume that $X$ is equipped with a hermitian metric $h$,
i.e.\ a positive definite hermitian form $h=\sum h_{jk}dz_j\otimes d\ol z_k$
of class $\cC^\infty$~; we denote by $\omega=\ii\sum h_{jk}dz_j\wedge d\ol z_k
\in \cC^\infty_{1,1}(X)$ the associated positive $(1,1)$-form.

\claim{(1.21) Definition} For every $T\in\cD^{\prime+}_{p,p}(X)$, the trace 
measure of $T$ with respect to $\omega$ is the positive measure
$$\sigma_T={1\over 2^pp!}\,T\wedge\omega^p.$$
\endclaim

If $(\zeta_1,\ldots,\zeta_n)$ is an orthonormal frame of $T^\star X$ with respect 
to $h$ on an open subset $U\subset X$, we may write
$$\eqalign{
&\omega=\ii\sum_{1\le j\le n}\zeta_j\wedge\ol\zeta_j,~~~~
\omega^p=i^{p^2}p!\,\sum_{|K|=p}\zeta_K\wedge\ol\zeta_K,\cr
&T=i^{(n-p)^2}\sum_{|I|=|J|=n-p}T_{I,J}\,\zeta_I\wedge\ol\zeta_J,~~~~
T_{I,J}\in\cD'(U),\cr}$$
where $\zeta_I=\zeta_{i_1}\wedge\ldots\wedge\zeta_{i_{n-p}}$. An easy 
computation yields
$$\sigma_T=2^{-p}\Big(\sum_{|I|=n-p}T_{I,I}\Big)\,\ii\zeta_1\wedge\ol\zeta_1
\wedge\ldots\wedge \ii\zeta_n\wedge\ol\zeta_n.\leqno(1.22)$$
For $X=\bC^n$ with the standard hermitian metric $h=\sum dz_j\otimes d\ol z_j$,
we get in particular
$$\sigma_T=2^{-p}\Big(\sum_{|I|=n-p}T_{I,I}\Big)\,
\ii dz_1\wedge d\ol z_1\wedge\ldots\wedge \ii dz_n\wedge d\ol z_n.
\leqno(1.22')$$
Proposition 1.14 shows that the mass measure $||T||=\sum|T_{I,J}|$ of
a positive current $T$ is always dominated by $C\sigma_T$ where $C>0$ is
a constant. It follows easily that the weak topology of
$\cD_p'(X)$ and of $\cD_p^{0\,\prime}(X)$ coincide on $\cD_p^{\prime+}(X)$,
which is moreover a metrizable subspace: its weak topology is in fact
defined by the collection of semi-norms $T\longmapsto
|\langle T,f_\nu\rangle|$ where $(f_\nu)$ is an arbitrary dense sequence
in $\cD_p(X)$. By the Banach-Alaoglu theorem, the unit ball in the dual of a
Banach space is weakly compact, thus:

\claim{(1.23) Proposition} Let $\delta$ be a positive continuous function
on $X$. Then the set of currents $T\in\cD_p^{\prime+}(X)$ such that
$\int_X \delta\,T\wedge\omega^p\le 1$ is weakly compact.
\endclaim

\proof{Proof} Note that our set is weakly closed, since a weak limit of
positive currents is positive and $\int_X \delta\,T\wedge\omega^p=
\sup\langle T,\theta\delta\omega^p\rangle$ when $\theta$ runs over
all elements of $\cD(X)$ such that $0\le\theta\le 1$.\qed
\endproof

Now, let $Z$ be a $p$-dimensional complex analytic submanifold of $X$.
We claim that
$$\sigma_{[Z]}={1\over 2^pp!}[Z]\wedge\omega^p={\rm Riemannian~volume~measure
~on~~}Z.\leqno(1.24)$$
This result is in fact a special case of the following important inequality.

\claim{(1.25) Wirtinger's inequality} Let $Y$ be an oriented
real submanifold of class $C^1$ and dimension $2p$ in $X$, and let $dV_Y$
be the Riemannian volume form on $Y$ associated with the metric 
$h_{\restriction Y}$. Set
$${1\over 2^pp!}\omega^p_{\restriction Y}=\alpha\,dV_Y,~~~~\alpha\in C^0(Y).$$
Then $|\alpha|\le 1$ and the equality holds if and only if $Y$ is a complex
analytic submanifold of $X$. In that case $\alpha=1$ if the orientation of
$Y$ is the canonical one, $\alpha=-1$ otherwise.
\endclaim

\proof{Proof} The restriction $\omega_{\restriction Y}$ is a real
antisymmetric $2$-form on $TY$. At any point $z\in Y$, we can thus find an
oriented orthonormal $\bR$-basis $(e_1,e_2,\ldots,e_{2p})$ of $T_zY$ such that
$$\eqalign{
{1\over 2}\omega&=\sum_{1\le k\le p}\alpha_k\,e^\star_{2k-1}\wedge
e^\star_{2k}~~~{\rm on}~~T_zY,~~~~{\rm where}\cr
\alpha_k&={1\over 2}\omega(e_{2k-1},e_{2k})=-\Im h(e_{2k-1},e_{2k}).\cr}$$
We have $dV_Y=e^\star_1\wedge\ldots\wedge e^\star_{2p}$ by definition of
the Riemannian volume form. By taking the $p$-th power of $\omega$, we get
$${1\over 2^pp!}\omega^p_{\restriction T_zY}=\alpha_1\ldots\alpha_p\,
e^\star_1\wedge\ldots\wedge e^\star_{2p}=\alpha_1\ldots\alpha_p\,dV_Y.$$
Since $(e_k)$ is an orthonormal $\bR$-basis, we have 
$\Re h(e_{2k-1},e_{2k})=0$, thus
$h(e_{2k-1},e_{2k})=-\ii\alpha_k$.  As $|e_{2k-1}|=|e_{2k}|=1$, we get
$$0\le|e_{2k}\pm Je_{2k-1}|^2=2\big(1\pm\Re h(Je_{2k-1},e_{2k})\big)
=2(1\pm\alpha_k).$$
Therefore
$$|\alpha_k|\le 1,~~~~|\alpha|=|\alpha_1\ldots\alpha_p|\le 1,$$
with equality if and only if $\alpha_k=\pm1$ for all $k$, i.e.\
$e_{2k}=\pm Je_{2k-1}$. In this case $T_zY\subset T_zX$ is a complex vector
subspace at every point $z\in Y$, thus $Y$ is complex analytic by Lemma~I.4.23.
Conversely, assume that $Y$ is a $\bC$-analytic submanifold and that
$(e_1,e_3,\ldots,e_{2p-1})$ is an orthonormal complex basis of $T_zY$. If
$e_{2k}:=Je_{2k-1}$, then $(e_1,\ldots,e_{2p})$ is an orthonormal $\bR$-basis
corresponding to the canonical orientation and
$${1\over 2}\omega_{\restriction Y}=
\sum_{1\le k\le p}e^\star_{2k-1}\wedge e^\star_{2k},~~~~
{1\over 2^pp!}\omega^p_{\restriction Y}=e^\star_1\wedge\ldots\wedge e^\star_{2p}
=dV_Y.\eqno\square$$
\endproof

Note that in the case of the standard hermitian metric $\omega$ on
$X=\bC^n$, the form $\omega=\ii\sum dz_j\wedge d\ol z_j=
d\big(\ii\sum z_j\,d\ol z_j\big)$ is globally exact.
Under this hypothesis, we are going to see that $\bC$-analytic 
submanifolds are always \hbox{\it minimal surfaces} for the 
Plateau problem, which consists in finding a compact subvariety $Y$ 
of minimal area with prescribed boundary~$\partial Y$.

\claim{(1.26) Theorem} Assume that the $(1,1)$-form $\omega$ is exact, say
$\omega=d\gamma$ with $\gamma\in \cC^\infty_1(X,\bR)$, and 
let $Y,Z\subset X$ be $(2p)$-dimensional oriented compact real submanifolds of 
class $C^1$ with boundary. If $\partial Y=\partial Z$ and $Z$ is complex
analytic, then
$${\rm Vol}(Y)\ge{\rm Vol(Z)}.$$
\endclaim

\proof{Proof} Write $\omega=d\gamma$. Wirtinger's inequality and Stokes' theorem
imply
$$\eqalignno{
{\rm Vol}(Y)&\ge{1\over 2^pp!}\Big|\int_Y\omega^p\Big|
={1\over 2^pp!}\Big|\int_Y d(\omega^{p-1}\wedge\gamma)\Big|
={1\over 2^pp!}\Big|\int_{\partial Y}\omega^{p-1}\wedge\gamma\Big|,\cr
{\rm Vol}(Z)&={1\over 2^pp!}\int_Z\omega^p={1\over 2^pp!}\int_{\partial Z}
\omega^{p-1}\wedge\gamma=\pm{1\over 2^pp!}\int_{\partial Y}
\omega^{p-1}\wedge\gamma.&\square\cr}$$
\endproof

\section{2}{Closed Positive Currents}
\subsection{2.A}{The Skoda-El Mir Extension Theorem}
We first prove the Skoda-El Mir extension theorem (\cite{Skoda 1982}, 
\cite{El Mir 1984}),
which shows in particular that a closed positive current defined in the
complement of an analytic set $E$ can be extended through~$E$ if (and
only~if) the mass of the current is locally finite near~$E$.
El Mir simplified Skoda's argument and showed that it is enough to assume
$E$ complete pluripolar. We follow here the exposition of Sibony's
survey article \cite{Sibony 1985}.

\claim{(2.1) Definition} A subset $E\subset X$ is said to be complete
pluripolar in $X$ if for every point $x_0\in X$ there exist a neighborhood
$\Omega$ of $x_0$ and a function $u\in\Psh(\Omega)\cap L^1_\loc(\Omega)$
such that $E\cap\Omega=\{z\in\Omega~;~u(z)=-\infty\}$.
\endclaim

Note that any closed analytic subset $A\subset X$ is complete pluripolar:
if $g_1=\ldots=g_N=0$ are holomorphic equations of $A$ on an open set
$\Omega\subset X$, we can take $u=\log(|g_1|^2+\ldots+|g_N|^2)$.

\claim{(2.2) Lemma} Let $E\subset X$ be a closed complete pluripolar set. If
$x_0\in X$ and $\Omega$ is a sufficiently small neighborhood of $x_0$, 
there exists:
\medskip
\item{\rm a)} a function $v\in\Psh(\Omega)\cap \cC^\infty(\Omega\ssm E)$ 
such that $v=-\infty$ on $E\cap\Omega~;$
\medskip
\item{\rm b)} an increasing sequence $v_k\in\Psh(\Omega)\cap \cC^\infty(\Omega)$,
$0\le v_k\le 1$, converging uniformly to $1$ on every compact subset of
$\Omega\ssm E$, such that $v_k=0$ on a neighborhood of $E\cap\Omega$.
\smallskip
\endclaim

\proof{Proof} Assume that $\Omega_0\compact X$ is a coordinate patch of $X$
containing $x_0$ and that $E\cap\Omega_0=\{z\in\Omega_0~;~u(z)=-\infty\}$,
$u\in\Psh(\Omega_0)$. In addition, we can achieve $u\le 0$
by shrinking $\Omega_0$ and subtracting a constant to $u$. Select a
convex increasing function $\chi\in \cC^\infty([0,1],\bR)$ such that $\chi(t)=0$ on
$[0,1/2]$ and $\chi(1)=1$. We set
$$u_k=\chi\big(\exp(u/k)\big).$$
Then $0\le u_k\le1$, $u_k$ is plurisubharmonic on $\Omega_0$,
$u_k=0$ in a neighborhood $\omega_k$ of $E\cap\Omega_0$ and $\lim u_k=1$
on $\Omega_0\ssm E$. Let $\Omega\compact\Omega_0$ be a neighborhood
of $x_0$, let $\delta_0=d(\Omega,\complement\Omega_0)$
and $\varepsilon_k\in{}]0,\delta_0[$ be such that
$\varepsilon_k<d(E\cap\ol\Omega,\ol\Omega\ssm\omega_k)$. Then
$$w_k:=\max_{j\le k}\{u_j\star\rho_{\varepsilon_j}\}\in\Psh(\Omega)\cap C^0
(\Omega),$$
$0\le w_k\le1$, $w_k=0$ on a neighborhood of $E\cap\Omega$ and $w_k$
is an increasing sequence converging to $1$ on $\Omega\ssm E$
(note that $w_k\ge u_k$). Hence, the convergence is uniform on every
compact subset of $\Omega\ssm E$ by Dini's lemma. We may
therefore choose a subsequence $w_{k_s}$ such that $w_{k_s}(z)\ge 1-2^{-s}$
on an increasing sequence of open sets $\Omega'_s$
with $\bigcup\Omega'_s=\Omega\ssm E$. Then
$$w(z):=|z|^2+\sum_{s=0}^{+\infty}(w_{k_s}(z)-1)$$
is a strictly plurisubharmonic function on $\Omega$ that is continuous
on $\Omega\ssm E$, and $w=-\infty$ on $E\cap\Omega$.
Richberg's theorem~I.3.40 applied on $\Omega\ssm E$ produces
$v\in\Psh(\Omega\ssm E)\cap \cC^\infty(\Omega\ssm E)$ such that
$w\le v\le w+1$. If we set $v=-\infty$ on $E\cap\Omega$, then $v$ is 
plurisubharmonic on $\Omega$ and has the properties required in a).
After subtraction of a constant, we may assume $v\le 0$ on $\Omega$.
Then the sequence $(v_k)$ of statement b) is obtained by letting
$v_k=\chi\big(\exp(v/k)\big)$.\qed
\endproof

\claim{(2.3) Theorem {\rm (El Mir)}} Let $E\subset X$ be a closed
complete pluripolar set and $T\in\cD^{\prime+}_{p,p}(X\ssm E)$ a
closed positive current. Assume that $T$ has finite mass in a
neighborhood of every point of $E$. Then the trivial extension
$\tilde T\in\cD^{\prime+}_{p,p}(X)$ obtained by extending the
measures $T_{I,J}$ by $0$ on $E$ is closed on $X$.
\endclaim

\proof{Proof} The statement is local on $X$, so we may work on a small
open set $\Omega$ such that there exists a sequence $v_k\in\Psh(\Omega)\cap
\cC^\infty(\Omega)$ as in 2.2~b). Let $\theta\in \cC^\infty([0,1])$ be a function
such that $\theta=0$ on $[0,1/3]$, $\theta=1$ on $[2/3,1]$ and 
$0\le\theta\le 1$. Then $\theta\circ v_k=0$ near $E\cap\Omega$ and 
$\theta\circ v_k=1$ for $k$ large on every fixed compact subset of 
$\Omega\ssm E$. Therefore 
$\tilde T=\lim_{k\to+\infty}(\theta\circ v_k)T$ and
$$d'\tilde T=\lim_{k\to+\infty}T\wedge d'(\theta\circ v_k)$$
in the weak topology of currents. It is therefore sufficient to check that 
$T\wedge d'(\theta\circ v_k)$ converges weakly to $0$ in 
$\cD'_{p-1,p}(\Omega)$ (note that $d''\tilde T$ is conjugate to $d'\tilde T$, 
thus $d''\tilde T$ will also vanish).

Assume first that $p=1$. Then $T\wedge d'(\theta\circ v_k)\in\cD'_{0,1}
(\Omega)$, and we have to show that
$$\langle T\wedge d'(\theta\circ v_k),\ol\alpha\rangle=\langle T,
\theta'(v_k)d'v_k\wedge\ol\alpha\rangle
\longrightarrow 0,~~~~\forall\alpha\in\cD_{1,0}(\Omega).$$
As $\gamma\longmapsto\langle T,\ii\gamma\wedge\ol\gamma\rangle$ is a
non-negative hermitian form on $\cD_{1,0}(\Omega)$, the  Cauchy-Schwarz 
inequality yields
$$\big|\langle T,\ii\beta\wedge\ol\gamma\rangle\big|^2\le
\langle T,\ii\beta\wedge\ol\beta\rangle~
\langle T,\ii\gamma\wedge\ol\gamma\rangle,~~~~
\forall\beta,\gamma\in\cD_{1,0}(\Omega).$$
Let $\psi\in\cD(\Omega)$, $0\le\psi\le 1$, be equal to $1$ in a neighborhood
of ${\rm Supp}\,\alpha$. We find
$$\big|\langle T,\theta'(v_k)d'v_k\wedge\ol\alpha\rangle\big|^2\le
\langle T,\psi \ii d'v_k\wedge d''v_k\rangle~
\langle T,\theta'(v_k)^2 \ii\alpha\wedge\ol\alpha\rangle.$$
By hypothesis $\int_{\Omega\ssm E}T\wedge \ii\alpha\wedge\ol\alpha<
+\infty$ and $\theta'(v_k)$ converges everywhere to $0$ on $\Omega$, thus
$\langle T,\theta'(v_k)^2 \ii\alpha\wedge\ol\alpha\rangle$ converges to $0$
by Lebesgue's dominated convergence theorem. On the other hand
$$\eqalign{
&\ii d'd''v_k^2=2v_k\,\ii d'd''v_k+2\ii d'v_k\wedge d''v_k\ge 2\ii d'v_k\wedge d''v_k,\cr
&2\langle T,\psi \ii d'v_k\wedge d''v_k\rangle\le\langle T,\psi \ii d'd''v_k^2
\rangle.\cr}$$
As $\psi\in\cD(\Omega)$, $v_k=0$ near $E$ and $d'T=d''T=0$ on $\Omega\ssm
E$, an integration by parts yields
$$\langle T,\psi \ii d'd''v_k^2\rangle=\langle T,v_k^2\ii d'd''\psi\rangle
\le C\int_{\Omega\ssm E}\|T\|<+\infty$$
where $C$ is a bound for the coefficients of $\psi$. Thus 
$\langle T,\psi \ii d'v_k\wedge d''v_k\rangle$ is bounded, 
and the proof is complete when $p=1$.

In the general case, let $\beta_s=\ii\beta_{s,1}\wedge\ol
\beta_{s,1}\wedge\ldots\wedge \ii\beta_{s,p-1}\wedge\ol
\beta_{s,p-1}$ be a basis of forms of bidegree $(p-1,p-1)$
with constant coefficients (Lemma~1.4). Then $T\wedge\beta_s
\in\cD^{\prime+}_{1,1}(\Omega\ssm E)$ has finite mass near $E$ and is 
closed on $\Omega\ssm E$. Therefore $d(\tilde T\wedge\beta_s)=
(d\tilde T)\wedge\beta_s=0$ on $\Omega$ for all $s$, and
we conclude that $d\tilde T=0$.\qed
\endproof

\claim{(2.4) Corollary} If $T\in\cD^{\prime+}_{p,p}(X)$ is closed,
if $E\subset X$ is a closed complete pluripolar set and $\bOne_E$ is
its characteristic function, then $\bOne_ET$ and $\bOne_{X\ssm E}T$ 
are closed $($and, of course, positive$)$.
\endclaim

\proof{Proof} If we set $\Theta=T_{\restriction X\ssm E}$, then $\Theta$ has
finite mass near $E$ and we have $\bOne_{X\ssm E}T=\tilde\Theta$ and
$\bOne_ET=T-\tilde\Theta$.\qed
\endproof

\subsection{2.B}{Current of Integration over an Analytic Set}
Let $A$ be a pure $p$-dimensional analytic subset of a complex manifold
$X$. We would like to generalize Example~1.20 and to define a current of
integration $[A]$ by letting
$$\langle[A],v\rangle=\int_{A_\reg}v,~~~~v\in\cD_{p,p}(X).\leqno(2.5)$$
One difficulty is of course to verify that the integral converges near
$A_\sing$. This follows from the following lemma, due to \cite{Lelong 1957}.

\claim{(2.6) Lemma} The current $[A_\reg]\in\cD^{\prime+}_{p,p}
(X\ssm A_\sing)$ has finite mass in a neighbor\-hood of every point 
$z_0\in A_\sing$.
\endclaim

\proof{Proof} Set $T=[A_\reg]$ and let $\Omega\ni z_0$ be a coordinate open set.
If we write the monomials $dz_K\wedge d\ol z_L$ in terms of an 
arbitrary basis of $\Lambda^{p,p}T^\star\Omega$ consisting of
decomposable forms $\beta_s=\ii\beta_{s,1}\wedge\ol\beta_{s,1}
\wedge\ldots\wedge\beta_{s,p}\wedge\ol\beta_{s,p}$ 
(cf.\ Lemma~1.4), we see that the measures $T_{I,J}\,.\,\tau$ are
linear combinations of the positive measures $T\wedge\beta_s$.
It is thus sufficient to prove that all $T\wedge\beta_s$
have finite mass near $A_\sing$. Without loss of generality, we may 
assume that $(A,z_0)$ is irreducible. Take new coordinates $w=(w_1,\ldots,w_n)$
such that $w_j=\beta_{s,j}(z-z_0)$, $1\le j\le p$.
After a slight perturbation of the $\beta_{s,j}$,
we may assume that each projection
$$\pi_s:A\cap(\Delta'\times\Delta''),~~~~w\longmapsto w'=(w_1,\ldots,w_p)$$
defines a ramified covering of $A$ (cf.\ Prop.~II.3.8 and Th.~II.3.19),
and that $(\beta_s)$ remains a basis of $\Lambda^{p,p}T^\star\Omega$.
Let $S$ be the ramification locus of $\pi_s$ and
$A_S=A\cap\big((\Delta'\ssm S)\times\Delta''\big)\subset A_\reg$.
The restriction of $\pi_s$: $A_S\longrightarrow\Delta'\ssm S$ is then a covering
with finite sheet number $q_s$ and we find
$$\eqalign{
&\int_{\Delta'\times\Delta''}[A_\reg]\wedge\beta_s=
\int_{A_\reg\cap(\Delta'\times\Delta'')}\ii dw_1\wedge d\ol w_1
\wedge\ldots\wedge \ii dw_p\wedge d\ol w_p\cr
&~~=\int_{A_S}\ii dw_1\wedge d\ol w_1\ldots\wedge d\ol w_p=
q_s\int_{\Delta'\ssm S}\ii dw_1\wedge d\ol w_1\ldots\wedge d\ol w_p<+\infty.\cr}$$
The second equality holds because $A_S$ is the complement in
$A_\reg\cap(\Delta'\times\Delta'')$ of an analytic subset
(such a set is of zero Lebesgue measure in $A_\reg$).\qed
\endproof

\claim{(2.7) Theorem {\rm (\cite{Lelong 1957})}} For every pure 
$p$-dimensional analytic subset $A\subset X$, the current of integration
$[A]\in\cD^{\prime+}_{p,p}(X)$ is a closed positive current on $X$.
\endclaim

\proof{Proof} Indeed, $[A_\reg]$ has finite mass near $A_\sing$
and $[A]$ is the trivial extension of $[A_\reg]$ to $X$ through the
complete pluripolar set $E=A_\sing$. Theorem~2.7 is then a 
consequence of El Mir's theorem.\qed
\endproof

\subsection{2.C}{Support Theorems and Lelong-Poincar\'e Equation}
Let $M\subset X$ be a closed $C^1$ real submanifold of $X$. The {\it
holomorphic tangent space} at a point $x\in M$ is
$${}^hT_xM=T_xM\cap JT_xM,\leqno(2.8)$$
that is, the largest complex subspace of $T_xX$ contained in $T_xM$.
We define the {\it Cauchy-Riemann dimension} of $M$ at $x$ by
${\rm CRdim}_xM=\dim_\bC {}^hT_xM$ and say that $M$ is a {\it CR submanifold}
of $X$ if ${\rm CRdim}_xM$ is a constant. In general, we set
$${\rm CRdim}~M=\max_{x\in M}~{\rm CRdim}_xM=
\max_{x\in M}~\dim_\bC {}^hT_xM.\leqno(2.9)$$
A current $\Theta$ is said to be {\it normal} if $\Theta$ and $d\Theta$
are currents of order $0$. For instance, every closed positive current
is normal. We are going to prove two important theorems
describing the structure of normal currents with support in $CR$
submanifolds.

\claim{(2.10) First theorem of support} Let $\Theta\in\cD'_{p,p}(X)$
be a normal current. If ${\rm Supp}\,\Theta$ is contained in a real 
submanifold $M$ of CR dimension $<p$, then $\Theta=0$.
\endclaim

\proof{Proof} Let $x_0\in M$ and let $g_1,\ldots,g_m$ be real $C^1$ functions in 
a neighbor\-hood $\Omega$ of $x_0$ such that $M=\{z\in\Omega~;~g_1(z)=\ldots=
g_m(z)=0\}$ and $dg_1\wedge\ldots\wedge dg_m\ne 0$ on $\Omega$. Then
$${}^hTM=TM\cap JTM=\bigcap_{1\le k\le m}\ker dg_k\cap\ker(dg_k\circ J)=
\bigcap_{1\le k\le m}\ker d'g_k$$
because $d'g_k={1\over 2}\big(dg_k-i(dg_k)\circ J\big)$. As $\dim_\bC
{}^hTM<p$, the rank of the system of $(1,0)$-forms $(d'g_k)$ must be
$>n-p$ at every point of $M\cap\Omega$. After a change of the 
ordering, we may assume for example that $\zeta_1=d'g_1$, $\zeta_2=d'g_2$,
$\ldots$, $\zeta_{n-p+1}=d'g_{n-p+1}$ are linearly independent on $\Omega$
(shrink $\Omega$ if necessary).
Complete $(\zeta_1,\ldots,\zeta_{n-p+1})$ into a continuous frame
$(\zeta_1,\ldots,\zeta_n)$ of $T^\star X_{\restriction\Omega}$ and set
$$\Theta=\sum_{|I|=|J|=n-p}\Theta_{I,J}\,\zeta_I\wedge\ol\zeta_J~~~{\rm on}~~
\Omega.$$
As $\Theta$ and $d'\Theta$ have measure coefficients supported on $M$
and $g_k=0$ on $M$, we get $g_k\Theta=g_kd'\Theta=0$, thus
$$d'g_k\wedge\Theta=d'(g_k\Theta)-g_kd'\Theta=0,~~~~1\le k\le m,$$
in particular $\zeta_k\wedge\Theta=0$ for all $1\le k\le n-p+1$.
When $|I|=n-p$, the multi-index $\complement I$ contains at least one of the 
elements $1,\ldots,n-p+1$, hence $\Theta\wedge\zeta_{\complement I}\wedge\ol
\zeta_{\complement J}=0$ and $\Theta_{I,J}=0$.\qed
\endproof

\claim{(2.11) Corollary} Let $\Theta\in\cD'_{p,p}(X)$ be a normal current.
If ${\rm Supp}\,\Theta$ is contained in an analytic subset $A$ of dimension
$<p$, then $\Theta=0$.
\endclaim

\proof{Proof} As $A_\reg$ is a submanifold of CRdim $<p$ in $X\ssm A_\sing$,
Theorem~2.9 shows that $\Supp\Theta\subset A_\sing$ and we conclude 
by induction on $\dim A$.\qed
\endproof

Now, assume that $M\subset X$ is a CR submanifold of class $C^1$
with CRdim$\,M=p$ and that ${}^hTM$ is an integrable subbundle of $TM\,$;
this means that the Lie bracket of two vector fields in ${}^hTM$ is in ${}^hTM$.
The Frobenius integrability theorem then shows that $M$ is locally fibered
by complex analytic $p$-dimensional submanifolds. More precisely, in a
neighborhood of every point of $M$, there is a submersion
$\sigma:M\longrightarrow Y$ onto a real $C^1$ manifold $Y$ such that the tangent
space to each fiber $F_t=\sigma^{-1}(t)$, $t\in Y$, is the holomorphic
tangent space ${}^hTM\,$; moreover, the fibers $F_t$ are necessarily 
complex analytic in view of Lemma 1.7.18. Under these assumptions,
with any complex measure $\mu$ on $Y$ we associate a current $\Theta$ 
with support in $M$ by
$$\Theta=\int_{t\in Y}[F_t]\,d\mu(t),~~~~
\hbox{\rm i.e.}~~\langle\Theta,u\rangle=\int_{t\in Y}
\Big(\int_{F_t}u\Big)\,d\mu(t)\leqno(2.12)$$
for all $u\in\cD'_{p,p}(X)$.  Then clearly $\Theta\in\cD'_{p,p}(X)$ 
is a closed current of order $0$, for all fibers $[F_t]$ have the same
properties.  When the fibers $F_t$ are connected, the following 
converse statement holds:

\claim{(2.13) Second theorem of support} Let $M\subset X$ be a CR
submanifold of CR dimension $p$ such that there is a submersion
$\sigma:M\longrightarrow Y$ of class $C^1$ whose fibers $F_t=\sigma^{-1}(t)$ are
connected and are the integral manifolds of the holomorphic tangent 
space ${}^hTM$.
Then any closed current $\Theta\in\cD'_{p,p}(X)$ of order $0$
with support in $M$ can be written $\Theta=\int_Y[F_t]\,d\mu(t)$ with
a unique complex measure $\mu$ on $Y$. Moreover $\Theta$ is $($strongly$)$ 
positive if and only if the measure $\mu$ is positive.
\endclaim

\proof{Proof} Fix a compact set $K\subset Y$ and a $C^1$ retraction $\rho$
from a neighborhood $V$ of $M$ onto $M$. By means of a partition of unity,
it is easy to construct a positive form $\alpha\in\cD_{p,p}^0(V)$ such 
that $\int_{F_t}\alpha=1$ for each fiber $F_t$ with $t\in K$. Then
the uniqueness and positivity statements for $\mu$ follow from the
obvious formula
$$\int_Y f(t)\,d\mu(t)=\langle\Theta,(f\circ\sigma\circ\rho)
\,\alpha\rangle,~~~~
\forall f\in C^0(Y),~~\Supp f\subset K.$$
Now, let us prove the existence of $\mu$. Let $x_0\in M$. 
There is a small neighborhood $\Omega$ of $x_0$
and real coordinates $(x_1,y_1,\ldots,x_p,y_p,t_1,\ldots,t_q,g_1,\ldots,g_m)$
such that
\medskip
\noindent $\bullet$ $z_j=x_j+\ii y_j$, $1\le j\le p$, are holomorphic
functions on $\Omega$ that define complex coordinates on all fibers
$F_t\cap\Omega$.
\smallskip
\noindent $\bullet$ $t_1,\ldots,t_q$ restricted to $M\cap\Omega$ are
pull-backs by $\sigma:M\to Y$ of local coordinates on an open set 
$U\subset Y$ such that $\sigma_{\restriction\Omega}:M\cap\Omega\longrightarrow U$ 
is a trivial fiber space.
\smallskip
\noindent $\bullet$ $g_1=\ldots=g_m=0$ are equations of $M$ in $\Omega$.
\medskip
\noindent Then $TF_t=\{dt_j=dg_k=0\}$ equals ${}^hTM=\{d'g_k=0\}$ and the 
rank of $(d'g_1,\ldots,d'g_m)$ is equal to $n-p$ at every point of $M\cap\Omega$.
After a change of the ordering we may suppose that $\zeta_1=d'g_1$, $\ldots$,
$\zeta_{n-p}=d'g_{n-p}$ are linearly independent on $\Omega$. As in
Prop.~2.10, we get $\zeta_k\wedge\Theta=\ol\zeta_k\wedge\Theta=0$
for $1\le k\le n-p$ and infer that $\Theta\wedge\zeta_{\complement I}
\wedge\ol\zeta_{\complement J}=0$ unless $I=J=L$ where
$L=\{1,2,\ldots,n-p\}$. Hence
$$\Theta=\Theta_{L,L}~\zeta_1\wedge\ldots\wedge\zeta_{n-p}\wedge
\ol\zeta_1\wedge\ldots\wedge\ol\zeta_{n-p}~~~~\hbox{\rm on}~~\Omega.$$
Now $\zeta_1\wedge\ldots\wedge\ol\zeta_{n-p}$ is proportional to
$dt_1\wedge\ldots dt_q\wedge dg_1\wedge\ldots\wedge dg_m$ because both
induce a volume form on the quotient space $TX_{\restriction M}/{}^hTM$.
Therefore, there is a complex measure $\nu$ supported on
$M\cap\Omega$ such that
$$\Theta=\nu~dt_1\wedge\ldots dt_q\wedge dg_1\wedge\ldots\wedge dg_m
~~~~\hbox{\rm on}~~\Omega.$$
As $\Theta$ is supposed to be closed, we have $\partial\nu/\partial x_j=
\partial\nu/\partial y_j=0$. Hence $\nu$ is a measure depending only
on $(t,g)$, with support in $g=0$. We may write $\nu=d\mu_U(t)\otimes
\delta_0(g)$ where $\mu_U$ is a measure on $U=\sigma(M\cap\Omega)$
and $\delta_0$ is the Dirac measure at $0$. If $j:M\longrightarrow X$ is the
injection, this means precisely that $\Theta=j_\star\sigma^\star\mu_U$
on $\Omega$, i.e.\
$$\Theta=\int_{t\in U}[F_t]\,d\mu_U(t)~~~~\hbox{\rm on}~~\Omega.$$
The uniqueness statement shows that for two open sets $\Omega_1$,
$\Omega_2$ as above, the associated measures $\mu_{U_1}$ and
$\mu_{U_2}$ coincide on $\sigma(M\cap\Omega_1\cap\Omega_2)$.
Since the fibers $F_t$ are connected, there is a unique measure $\mu$
which coincides with all measures $\mu_U$.\qed
\endproof

\claim{(2.14) Corollary} Let $A$ be an analytic subset of $X$ with global
irreducible components $A_j$ of pure dimension~$p$. Then any
closed current $\Theta\in\cD'_{p,p}(X)$ of order $0$ with support in $A$
is of the form $\Theta=\sum\lambda_j[A_j]$ where $\lambda_j\in\bC$.
Moreover, $\Theta$ is $($strongly$)$ positive if and only if all
coefficients $\lambda_j$ are $\ge 0$.
\endclaim

\proof{Proof} The regular part $M=A_\reg$ is a complex submanifold of $X\ssm
A_\sing$ and its connected components are $A_j\cap A_\reg$. Thus,
we may apply Th.~2.13 in the case where $Y$ is discrete to see
that $\Theta=\sum\lambda_j[A_j]$ on $X\ssm A_\sing$. Now
$\dim A_\sing<p$ and the
difference $\Theta-\sum\lambda_j[A_j]\in\cD'_{p,p}(X)$ is a closed 
current of order $0$ with support in $A_{\rm sing}$, so this current 
must vanish by Cor.~2.11.\qed
\endproof

\claim{(2.15) Lelong-Poincar\'e equation} Let $f\in\cM(X)$ 
be a meromorphic function which does not vanish identically 
on any connected component of $X$ and let $\sum m_j Z_j$ be the divisor
of~$f$. Then the function $\log|f|$ is locally integrable on~$X$
and satisfies the equation
$${\ii\over\pi}d'd''\log|f|=\sum~m_j\,[Z_j]$$
in the space $\cD'_{n-1,n-1}(X)$ of currents of
bidimension $(n-1,n-1)$.
\endclaim

We refer to Sect. 2.6 for the definition of divisors, and especially to
(2.6.14). Observe that if $f$ is holomorphic, then $\log|f|\in\Psh(X)$,
the coefficients $m_j$ are positive integers and the right hand side
is a positive current in $\cD^{\prime+}_{n-1,n-1}(X)$.

\proof{Proof} Let $Z=\bigcup Z_j$ be the support of $\div(f)$.
Observe that the sum in the right hand side is locally finite
and that $d'd''\log|f|$ is supported on $Z$, since 
$$d'\log|f|^2=d'\log(f\ol f)={\ol f\,df\over f\ol f}={df\over f}~~~{\rm on}
~~X\ssm Z.$$
In a neighborhood $\Omega$ of a point $a\in Z_j\cap Z_\reg$, we can find
local coordinates $(w_1,\ldots,w_n)$ such that $Z_j\cap\Omega$ is given by
the equation $w_1=0$. Then Th.~2.6.6 shows that $f$ can be written
$f(w)=u(w)w_1^{m_j}$ with an invertible holomorphic function $u$
on a smaller neighborhood $\Omega'\subset\Omega$. Then we have
$$\ii d'd''\log|f|=\ii d'd''\big(\log|u|+m_j\log|w_1|\big)=m_j\,\ii 
d'd''\log|w_1|.$$
For $z\in\bC$, Cor.~I.3.4 implies
$$\ii d'd''\log|z|^2=-\ii d''\Big({dz\over z}\Big)=
-\ii\pi\delta_0\,d\ol z\wedge dz=2\pi\,[0].$$
If $\Phi:\bC^n\longrightarrow\bC$ is the projection $z\longmapsto z_1$ and
$H\subset\bC^n$
the hyperplane $\{z_1=0\}$, formula (1.2.19) shows that
$$\ii d'd''\log|z_1|=\ii d'd''\log|\Phi(z)|=\Phi^\star(\ii d'd''\log|z|)=
\pi\Phi^\star([0])=\pi\,[H],$$
because $\Phi$ is a submersion. We get therefore
${\ii\over\pi}d'd''\log|f|=m_j[Z_j]$ in $\Omega'$.
This implies that the Lelong-Poincar\'e equation is valid at least on 
$X\ssm Z_\sing$. As $\dim Z_\sing<n-1$, Cor.~2.11 shows that
the equation holds everywhere on~$X$.\qed
\endproof

\section{3}{Definition of Monge-Amp\`ere Operators}
Let $X$ be a $n$-dimensional complex manifold. We denote by $d=d'+d''$
the usual decomposition of the exterior derivative in terms of its $(1,0)$
and $(0,1)$ parts, and we set
$$d^c={1\over2\ii\pi}(d'-d'').$$
It follows in particular that $d^c$ is a real operator, i.e.\ $\ol{d^cu}=
d^c\ol u$, and that $dd^c={\ii\over\pi}d'd''$. Although not quite standard,
the $1/2\ii\pi$ normalization is very convenient for many purposes, since
we may then forget the factor $2\pi$ almost everywhere (e.g.\ in the
Lelong-Poincar\'e equation (2.15)). In this context, we have the
following integration by part formula.

\claim{(3.1) Formula} Let $\Omega\compact X$ be a smoothly bounded open
set in $X$ and let $f,g$ be forms of class $C^2$ on $\smash{\ol\Omega}$ of
pure bidegrees $(p,p)$ and $(q,q)$ with~\hbox{$p+q=n-1$}. Then
$$\int_\Omega f\wedge dd^c g-dd^c f\wedge g=
\int_{\partial\Omega} f\wedge d^c g-d^c f\wedge g.$$
\endclaim

\proof{Proof} By Stokes' theorem the right hand side is the integral over $\Omega$ of
$$d(f\wedge d^c g-d^c f\wedge g)= f\wedge dd^c g-dd^c f\wedge g+
(df\wedge d^c g+d^c f\wedge dg).$$
As all forms of total degree $2n$ and bidegree $\ne(n,n)$ are zero, we get
$$df\wedge d^cg={1\over 2\ii\pi}(d''f\wedge d'g-d'f\wedge d''g)=-d^cf\wedge
dg.\eqno{\square}$$
\endproof

Let $u$ be a plurisubharmonic function on $X$ and let $T$ be a closed
positive current of bidimension $(p,p)$, i.e.\ of bidegree $(n-p,n-p)$. 
Our desire is to define the wedge product $dd^cu\wedge T$ even when
neither $u$ nor $T$ are smooth.  A priori, this product does not make
sense because $dd^cu$ and $T$ have measure coefficients and measures
cannot be multiplied; see \cite{Kiselman 1983} for interesting
counterexamples. Assume however that $u$ is a {\it locally
bounded} plurisubharmonic function. Then the current $uT$ is well
defined since $u$ is a locally bounded Borel function and $T$ has
measure coefficients.  According to \cite{Bedford-Taylor 1982} we 
define
    $$dd^cu\wedge T=dd^c(uT)$$
where $dd^c(~~)$ is taken in the sense of distribution (or current) theory.

\claim{(3.2) Proposition} The wedge product $dd^c u\wedge T$ is 
again a closed positive current.
\endclaim

\proof{Proof} The result is local. In an open set $\Omega\subset\bC^n$,
we can use convolution with a family of regularizing kernels to
find a decreasing sequence of smooth plurisubharmonic
functions $u_k=u\star\rho_{1/k}$ converging pointwise to $u$.
Then $u\le u_k\le u_1$ and Lebesgue's dominated 
convergence theorem shows that $u_k T$ converges weakly to $uT\,$;
thus $dd^c(u_k T)$ converges weakly to $dd^c(u T)$ by the weak
continuity of differentiations. However, since $u_k$ is smooth,
$dd^c(u_k T)$ coincides with the product $dd^c u_k\wedge T$ in its usual
sense. As $T\ge0$ and as $dd^cu_k$ is a positive $(1,1)$-form, we have 
$dd^c u_k\wedge T\ge 0$, hence the weak limit $dd^c u\wedge T$ is 
$\ge 0$ (and obviously closed).\qed
\endproof

Given locally bounded plurisubharmonic functions $u_1,\ldots,u_q$, we define
inductively
$$dd^c u_1\wedge dd^cu_2\wedge\ldots\wedge dd^c u_q\wedge T=
dd^c(u_1dd^c u_2\wedge\ldots\wedge dd^c u_q\wedge T).$$
By (3.2) the product is a closed positive current. In particular, when $u$ is
a locally bounded plurisubharmonic function, the bidegree $(n,n)$ current
$(dd^cu)^n$ is well defined and is a positive measure. If $u$ is of class $C^2$,
a computation in local coordinates gives
$$(dd^cu)^n=\det\big({\partial^2u\over\partial z_j\partial\ol z_k}\Big)
\cdot{n!\over\pi^n}\,\ii dz_1\wedge d\ol z_1\wedge\ldots\wedge
\ii dz_n\wedge d\ol z_n.$$
The expression ``Monge-Amp\`ere operator" classically refers to the
non-linear partial differential operator
$u\longmapsto\det(\partial^2u/\partial z_j\partial\ol z_k)$. By extension,
all operators $(dd^c)^q$ defined above are also called Monge-Amp\`ere
operators.


Now, let $\Theta$ be a current of order $0$. When $K\compact X$ is 
an arbitrary compact subset, we define a {\it mass} semi-norm
$$||\Theta||_K=\sum_j\int_{K_j}~~\sum_{I,J}|\Theta_{I,J}|$$
by taking a partition $K=\bigcup K_j$ where each $\smash{\ol K_j}$ 
is contained in a coordinate patch and where $\Theta_{I,J}$ are the
corresponding measure coefficients. Up to constants, the semi-norm 
$||\Theta||_K$ does not depend on the choice of the coordinate systems 
involved. When $K$ itself is contained in a coordinate patch, we set
$\beta=dd^c|z|^2$ over $K\,$; then, if $\Theta\ge 0$, there are constants
$C_1,C_2>0$ such that
$$C_1||\Theta||_K\le\int_K\Theta\wedge\beta^p\le C_2||\Theta||_K.$$
We denote by $L^1(K)$, resp.\ by $L^\infty(K)$, the space of integrable
(resp.\ bounded measurable) functions on $K$ with respect to any smooth
positive density on~$X$.

\claim{(3.3) Chern-Levine-Nirenberg inequalities {\rm(1969)}} For 
all compact subsets
$K,L$ of $X$ with $L\subset K^\circ$, there exists a constant $C_{K,L}\ge0$ 
such that
$$||dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge T||_L\le C_{K,L}~
||u_1||_{L^\infty(K)}\ldots||u_q||_{L^\infty(K)}\,||T||_K.$$
\endclaim

\proof{Proof} By induction, it is sufficient to prove the result for $q=1$ and
$u_1=u$. There is a covering of $L$ by a family of balls
$B'_j\compact B_j\subset K$ contained in coordinate patches of $X$.
Let $\chi\in\cD(B_j)$ be equal to $1$ on $\smash{\ol B'_j}$. Then
$$||dd^cu\wedge T||_{L\cap\ol B'_j}\le C\int_{\ol B'_j}dd^c u\wedge T
\wedge\beta^{p-1}\le C\int_{B_j}\chi\,dd^c u\wedge T\wedge\beta^{p-1}.$$
As $T$ and $\beta$ are closed, an integration by parts yields
$$||dd^cu\wedge T||_{L\cap\ol B'_j}\le C\int_{B_j}u\,T\wedge dd^c\chi
\wedge\beta^{p-1}\le C'||u||_{L^\infty(K)}||T||_K$$
where $C'$ is equal to $C$ multiplied by a bound for the coefficients
of the smooth form $dd^c\chi\wedge\beta^{p-1}$.\qed
\endproof

\claim{(3.4) Remark} \rm With the same notations as above, any plurisubharmonic
function $V$ on $X$ satisfies inequalities of the type
\medskip
\item{\rm a)} $\quad||dd^cV||_L\le C_{K,L}\,||V||_{L^1(K)}$.
\medskip
\item{\rm b)} $\quad{\displaystyle\sup_L}\,V_+\le C_{K,L}\,||V||_{L^1(K)}$.
\medskip
\noindent In fact the inequality
$$\int_{L\cap\ol B'_j}dd^cV\wedge\beta^{n-1}\le\int_{B_j}\chi dd^cV\wedge
\beta^{n-1}=\int_{B_j}Vdd^c\chi\wedge\beta^{n-1}$$
implies a), and b) follows from the mean value inequality.
\endclaim

\claim{(3.5) Remark} \rm Products of the form $\Theta=\gamma_1\wedge\ldots\wedge
\gamma_q\wedge T$ with mixed $(1,1)$-forms $\gamma_j=dd^cu_j$ or 
$\gamma_j=dv_j\wedge d^cw_j+dw_j\wedge d^cv_j$ are also well defined 
whenever $u_j$, $v_j$, $w_j$ are locally bounded plurisubharmonic functions.
Moreover, for $L\subset K^\circ$, we have
$$||\Theta||_L\le C_{K,L}||T||_K\prod||u_j||_{L^\infty(K)}
\prod||v_j||_{L^\infty(K)}\prod||w_j||_{L^\infty(K)}.$$
To check this, we may suppose $v_j,w_j\ge 0$ and $||v_j||=||w_j||=1$ 
in~$L^\infty(K)$. Then the inequality 
follows from (3.3) by the polarization identity
$$2(dv_j\wedge d^cw_j+dw_j\wedge d^cv_j)=dd^c(v_j+w_j)^2
-dd^cv_j^2-dd^cw_j^2-v_jdd^cw_j-w_jdd^cv_j$$
in which all $dd^c$ operators act on plurisubharmonic functions.
\endclaim

\claim{(3.6) Corollary} Let $u_1,\ldots,u_q$ be continuous $($finite$)$ plurisubharmonic
functions and let $u_1^k,\ldots,u_q^k$ be sequences of plurisubharmonic functions
converging locally uniformly to $u_1,\ldots,u_q$. If $T_k$ is a sequence of closed
positive currents converging weakly to $T$, then
\smallskip
\item{\rm a)} $u_1^k dd^cu_2^k\wedge\ldots\wedge dd^cu_q^k
\wedge T_k\longrightarrow u_1dd^cu_2\wedge\ldots\wedge dd^cu_q\wedge T~~~{\it weakly}$.
\smallskip
\item{\rm b)} $dd^cu_1^k\wedge\ldots\wedge dd^cu_q^k\wedge T_k\longrightarrow
dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge T~~~{\it weakly}$.
\endclaim

\proof{Proof} We observe that b) is an immediate consequence of a) by the weak
continuity of $dd^c$. By using induction on $q$, it is enough to prove result
a) when~$q=1$. If $(u^k)$ converges locally uniformly to a finite
continuous plurisubharmonic function $u$, we introduce local regularizations
$u_\varepsilon=u\star\rho_\varepsilon$ and write
$$u^kT_k-uT=(u^k-u)T_k+(u-u_\varepsilon)T_k+u_\varepsilon(T_k-T).$$
As the sequence $T_k$ is weakly convergent, it is locally uniformly bounded
in mass, thus $||(u^k-u)T_k||_K\le ||u^k-u||_{L^\infty(K)}||T_k||_K$ converges
to $0$ on every compact set~$K$. The same argument shows that
$||(u-u_\varepsilon)T_k||_K$ can be made arbitrarily small by choosing
$\varepsilon$ small enough. Finally $u_\varepsilon$ is smooth, so
$u_\varepsilon(T_k-T)$ converges weakly to~$0$.\qed
\endproof

Now, we prove a deeper monotone continuity theorem due to
\cite{Bedford-Taylor 1982} according to which the
continuity and uniform convergence assumptions can be dropped if
each sequence $(u_j^k)$ is decreasing and $T_k$ is a constant sequence.

\claim{(3.7) Theorem} Let $u_1,\ldots,u_q$ be locally bounded
plurisubharmonic functions and let $u_1^k,\ldots,u_q^k$ be decreasing
sequences of plurisubharmonic functions converging pointwise to
$u_1,\ldots,u_q$. Then
\medskip
\item{\rm a)} $u_1^k dd^cu_2^k\wedge\ldots\wedge dd^cu_q^k
\wedge T\longrightarrow u_1dd^cu_2\wedge\ldots\wedge dd^cu_q\wedge T
~~~{\it weakly}$.
\medskip
\item{\rm b)} $dd^cu_1^k\wedge\ldots\wedge dd^cu_q^k\wedge T\longrightarrow
dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge T~~~{\it weakly}$.
\endclaim

\proof{Proof} Again by induction, observing that a)~$\Longrightarrow$~b)
and that a) is obvious for $q=1$ thanks to Lebesgue's bounded
convergence theorem.  To proceed with the induction step, we first
have to make some slight modifications of our functions $u_j$ and~$u_j^k$.

As the sequence $\smash{(u_j^k)}$ is decreasing and as $u_j$ is
locally bounded, the family $\smash{(u_j^k)_{k\in\bN}}$ is locally
uniformly bounded. The results are local, so we can work on a Stein open
set $\Omega\compact X$ with strongly pseudoconvex boundary. We use the
following notations:
\smallskip
{\parindent=12.48mm
\item{\swapleft{4pt}{(3.8)}}
let $\psi$ be a strongly plurisubharmonic function of class $\cC^\infty$ near
$\smash{\ol\Omega}$ with $\psi<0$ on $\Omega$ and $\psi=0$, $d\psi\ne 0$
on~$\partial\Omega\,;$
\smallskip
\item{\swapleft{4pt}{$(3.8')$}}
we set $\Omega_\delta=\{z\in\Omega\,;\,\psi(z)<-\delta\}$ for all $\delta>0$.
\vskip0pt}

\vbox{%
\input epsfiles/fig_3_1.tex
\vskip6mm
\centerline{{\bf III-1} Construction of $v_j^k$}}
\vskip5mm

\noindent After addition of a constant we can
assume that $-M\le\smash{u_j^k}\le-1$ near~$\smash{\ol\Omega}$. Let us denote
by $\smash{(u_j^{k,\varepsilon})}$, $\varepsilon\in{}]0,\varepsilon_0]$, an
increasing family of regularizations converging to $\smash{u_j^k}$ as 
$\varepsilon\to 0$ and such that $-M\le\smash{u^{k,\varepsilon}_j}\le-1$
on $\smash{\ol\Omega}$. Set $A=M/\delta$ with $\delta>0$ small and replace
$\smash{u_j^k}$ by $\smash{v_j^k=\max\{A\psi,u_j^k\}}$ and
$\smash{u_j^{k,\varepsilon}}$ by $\smash{v_j^{k,\varepsilon}=
\max_\varepsilon\{A\psi,u_j^{k,\varepsilon}\}}$
where $\max_\varepsilon=\max~\star~\rho_\varepsilon$ is a regularized 
max function.

Then $\smash{v_j^k}$ coincides with $\smash{u_j^k}$ on 
$\Omega_\delta$ since $A\psi<-A\delta=-M$ on
$\Omega_\delta$, and $\smash{v_j^k}$ is equal to 
$A\psi$ on the corona $\Omega\setminus\Omega_{\delta/M}$.  Without loss 
of generality, we can therefore assume that all $\smash{u_j^k}$ 
(and similarly all $\smash{u_j^{k,\varepsilon}})$ coincide with $A\psi$ 
on a fixed neighborhood of $\partial\Omega$. We need a lemma.
\endproof

\claim{(3.9) Lemma} Let $f_k$ be a decreasing sequence of upper
semi-continuous functions conver\-ging to $f$ on some separable locally
compact space $X$ and $\mu_k$ a sequence of positive measures
converging weakly to $\mu$ on $X$. Then every weak limit $\nu$ of $f_k\mu_k$
satisfies $\nu\le f\mu$.
\endclaim

Indeed if $(g_p)$ is a decreasing sequence of continuous functions
converging to $f_{k_0}$ for some $k_0$, then $f_k\mu_k\le f_{k_0}\mu_k
\le g_p\mu_k$ for $k\ge k_0$, thus $\nu\le g_p\mu$ as $k\to+\infty$.
The monotone convergence theorem then gives $\nu\le f_{k_0}\mu$ as
$p\to+\infty$ and $\nu\le f\mu$ as $k_0\to+\infty$.\qed

\proof{Proof of Theorem 3.7 (end).} Assume that a) has been proved
for $q-1$. Then 
$$S^k=dd^cu_2^k\wedge\ldots\wedge dd^cu_q^k\wedge T\longrightarrow 
S=dd^cu_2\wedge\ldots\wedge dd^cu_q\wedge T.$$ 
By 3.3 the sequence $(u_1^k S^k)$ has locally bounded mass, hence is 
relatively compact for the weak topology.  In order to prove a), we only
have to show that every weak limit $\Theta$ of $u_1^k S^k$ is equal to 
$u_1 S$.  Let $(m,m)$ be the bidimension of $S$ and let $\gamma$ be an
arbitrary smooth and strongly positive~form of bidegree $(m,m)$.  Then
the positive measures $S^k\wedge\gamma$ converge weakly to
$S\wedge\gamma$ and Lemma~3.9 shows that $\Theta\wedge\gamma\le
u_1S\wedge\gamma$, hence $\Theta\le u_1S$.  To get~the equality, we set
$\beta=dd^c\psi>0$ and show that
$\int_\Omega u_1 S\wedge\beta^m\le\int_\Omega\Theta\wedge\beta^m$, i.e.\ 
$$\int_\Omega u_1dd^cu_2\wedge\ldots\wedge dd^cu_q\wedge T\wedge\beta^m\le
\liminf_{k\to+\infty}\int_\Omega u_1^k dd^cu_2^k\wedge\ldots\wedge dd^cu_q^k
\wedge T\wedge\beta^m.$$ 
As $u_1\le u_1^k\le u_1^{k,\varepsilon_1}$ for every $\varepsilon_1>0$, we get
$$\eqalign{\int_\Omega u_1&dd^cu_2\wedge\ldots\wedge dd^cu_q\wedge
T\wedge\beta^m\cr
&\le\int_\Omega u_1^{k,\varepsilon_1}dd^cu_2
\wedge\ldots\wedge dd^cu_q\wedge T\wedge\beta^m\cr 
&=\int_\Omega dd^c u_1^{k,\varepsilon_1}\wedge u_2dd^cu_3
\wedge\ldots\wedge dd^cu_q\wedge T\wedge\beta^m\cr}$$ 
after an integration by parts (there is no boundary term because
$\smash{u_1^{k,\varepsilon_1}}$ and $u_2$ both vanish on $\partial\Omega$).  
Repeating this argument with $u_2,\ldots,u_q$, we obtain
$$\eqalign{
\int_\Omega u_1&dd^cu_2\wedge\ldots\wedge dd^cu_q\wedge T\wedge\beta^m\cr 
&\le\int_\Omega dd^c u_1^{k,\varepsilon_1}\wedge\ldots\wedge dd^c
u_{q-1}^{k,\varepsilon_{q-1}}\wedge u_q T\wedge\beta^m\cr
&\le\int_\Omega u_1^{k,\varepsilon_1}dd^cu_2^{k,\varepsilon_2}
\wedge\ldots\wedge dd^c u_q^{k,\varepsilon_q}\wedge T\wedge\beta^m.\cr}$$ 
Now let $\varepsilon_q\to 0,\ldots,\varepsilon_1\to 0$ in this order.  We
have weak convergence at each step and $\smash{u_1^{k,\varepsilon_1}}=0$
on the boundary; therefore the integral in the last line converges and we
get the desired inequality 
$$\int_\Omega u_1dd^cu_2\wedge\ldots\wedge dd^cu_q\wedge
T\wedge\beta^m\le\int_\Omega u_1^k dd^cu_2^k\wedge\ldots\wedge
dd^cu_q^k\wedge T\wedge\beta^m.\eqno{\square}$$
\endproof

\claim{(3.10) Corollary} The product $dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge T$
is symmetric with respect to $u_1,\ldots,u_q$.
\endclaim

\proof{Proof} Observe that the definition was unsymmetric. The
result is true when $u_1,\ldots,u_q$ are smooth and follows in general from
Th.~3.7 applied to the sequences $u_j^k=u_j\star\rho_{1/k}$,
$1\le j\le q$.\qed
\endproof

\claim{(3.11) Proposition} Let $K,L$ be compact subsets of $X$ such that
$L\subset K^\circ$. For any plurisubharmonic functions $V,u_1,\ldots,u_q$ on $X$ 
such that $u_1,\ldots,u_q$ are locally bounded, there is an inequality
$$||Vdd^cu_1\wedge\ldots\wedge dd^cu_q||_L\le 
C_{K,L}\,||V||_{L^1(K)}||u_1||_{L^\infty(K)}\ldots||u_q||_{L^\infty(K)}.$$
\endclaim

\proof{Proof} We may assume that $L$ is contained in a strongly pseudoconvex open
set $\Omega=\{\psi<0\}\subset K$ (otherwise we cover $L$ by
small balls contained in $K$).  A suitable normalization gives
$-2\le u_j\le -1$ on $K\,$;  then we can modify $u_j$ on 
$\Omega\setminus L$ so that $u_j=A\psi$ on
$\Omega\setminus\Omega_\delta$ with a fixed constant $A$ and
$\delta>0$ such that $L\subset\Omega_\delta$.  Let $\chi\ge 0$
be a smooth function equal to $-\psi$ on $\Omega_\delta$ with compact
support in $\Omega$.  If we take $||V||_{L^1(K)}=1$, we see that $V_+$
is uniformly bounded on $\Omega_\delta$ by 3.4~b); after subtraction
of a fixed constant we can assume $V\le 0$ on $\Omega_\delta$.  First
suppose $q\le n-1$. As $u_j=A\psi$ on $\Omega\setminus\Omega_\delta$,
we find
$$\eqalign{
&\int_{\Omega_\delta}-V\,dd^cu_1\wedge\ldots\wedge
dd^cu_q\wedge\beta^{n-q}\cr
&~\,{}=\int_\Omega V\,dd^cu_1\wedge\ldots\wedge
dd^cu_q\wedge\beta^{n-q-1}\wedge dd^c\chi-A^q\int_{\Omega\setminus
\Omega_\delta} V\,\beta^{n-1}\wedge dd^c\chi\cr
&~\,{}=\int_\Omega\chi\,dd^cV\wedge dd^cu_1\wedge\ldots\wedge
dd^cu_q\wedge\beta^{n-q-1}-A^q\int_{\Omega\setminus
\Omega_\delta} V\,\beta^{n-1}\wedge dd^c\chi.\cr}$$
The first integral of the last line is uniformly bounded thanks to
3.3 and 3.4~a), and the second one is bounded by $||V||_{L^1(\Omega)}\le$
constant. Inequality 3.11 follows for $q\le n-1$. If $q=n$,
we can work instead on $X\times\bC$ and consider $V,u_1,\ldots,u_q$ as
functions on $X\times\bC$ independent of the extra variable
in~$\bC$.\qed
\endproof{}

\section{4}{Case of Unbounded Plurisubharmonic Functions}
We would like to define $dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge T$
also in some cases when $u_1,\ldots,u_q$ are not bounded below everywhere,
especially when the $u_j$ have logarithmic poles.
Consider first the case $q=1$ and let $u$ be a plurisubharmonic function
on $X$.  The {\it pole set} of $u$ is by definition $P(u)=u^{-1}(-\infty)$.
We define the {\it unbounded locus} $L(u)$ to be the set of points $x\in X$
such that $u$ is unbounded in every neighborhood of~$x$. Clearly $L(u)$ is
closed and we have $L(u)\supset\smash{\ol{P(u)}}$ but in general
these sets are different: in fact, $u(z)=\sum k^{-2}\log(|z-1/k|+e^{-k^3})$
is everywhere finite in $\bC$ but $L(u)=\{0\}$.

\claim{(4.1) Proposition} We make two additional assumptions:
\smallskip
\item{\rm a)} $T$ has non zero bidimension $(p,p)$ (i.e.\ degree of $T<2n$).
\smallskip
\item{\rm b)} $X$ is covered by a family of Stein open
sets $\Omega\compact X$ whose boundaries $\partial\Omega$ do not
intersect $L(u)\cap\Supp\,T$.
\smallskip
\noindent Then the current $uT$ has locally finite mass in X.
\endclaim

For any current $T$, hypothesis 4.1~b) is clearly satisfied 
when $u$ has a discrete unbounded locus $L(u)$; an interesting example is
$u=\log|F|$ where $F=(F_1,\ldots,F_N)$ are holomorphic functions having
a discrete set of common zeros.
Observe that the current $uT$ need not have locally finite
mass when $T$ has degree $2n$ (i.e.\ $T$ is a measure); example:
$T=\delta_0$ and $u(z)=\log|z|$ in $\bC^n$. The result also fails
when the sets $\Omega$ are not assumed to be Stein; example: $X={}$
blow-up of $\bC^n$ at $0$, $T=[E]={}$ current of integration on the
exceptional divisor and $u(z)=\log|z|$ (see \S~7.12 for the definition
of blow-ups).

\proof{Proof} By shrinking $\Omega$ slightly, we may assume that $\Omega$ has a
smooth strongly pseudoconvex boundary. Let $\psi$ be a defining
function of $\Omega$ as in (3.8). By subtracting a constant to $u$, we may
assume $u\le-\varepsilon$ on $\ol\Omega$. We fix $\delta$ so small that
$\ol\Omega\ssm\Omega_\delta$ does not intersect $L(u)\cap\Supp\,T$ and
we select a neighborhood $\omega$ of $(\ol\Omega\ssm\Omega_\delta)\cap
\Supp\,T$ such that $\ol\omega\cap L(u)=\emptyset$. Then we define
$$u_s(z)=\cases{
\max\{u(z),A\psi(z)\}&on $\omega$,\cr
\max\{u(z),s\}&on $\Omega_\delta=\{\psi<-\delta\}$.\cr}$$
By construction $u\ge -M$ on $\omega$ for some constant $M>0$. We fix
$A\ge M/\delta$ and take $s\le -M$, so
$$\max\{u(z),A\psi(z)\}=\max\{u(z),s\}=u(z)~~~~\hbox{\rm on}~~
\omega\cap\Omega_\delta$$
and our definition of $u_s$ is coherent. Observe that $u_s$ is defined
on $\omega\cup\Omega_\delta$, which is a neighborhood of
$\ol\Omega\cap\Supp\,T$. Now, $u_s=A\psi$ on
$\omega\cap(\Omega\ssm\Omega_{\varepsilon/A})$, hence Stokes' theorem implies
$$\eqalign{
\int_\Omega dd^cu_s\wedge T\wedge(dd^c\psi)^{p-1}&{}-
\int_\Omega Add^c\psi\wedge T\wedge(dd^c\psi)^{p-1}\cr
&=\int_\Omega dd^c\big[(u_s-A\psi)T\wedge(dd^c\psi)^{p-1}\big]=0\cr}$$
because the current $[\ldots]$ has a compact support contained in
$\ol\Omega_{\varepsilon/A}$. Since $u_s$ and $\psi$ both vanish on
$\partial\Omega$, an integration by parts gives
$$\eqalign{
\int_\Omega u_sT\wedge(dd^c\psi)^p
&=\int_\Omega\psi dd^cu_s\wedge T\wedge(dd^c\psi)^{p-1}\cr
&\ge-||\psi||_{L^\infty(\Omega)}\int_\Omega T\wedge dd^cu_s\wedge
(dd^c\psi)^{p-1}\cr
&=-||\psi||_{L^\infty(\Omega)}A\int_\Omega T\wedge(dd^c\psi)^p.\cr}$$
Finally, take $A=M/\delta$, let $s$ tend to $-\infty$ and use the inequality
$u\ge-M$ on~$\omega$. We obtain
$$\eqalign{
\int_\Omega u\,T\wedge(dd^c\psi)^p
&\ge -M\int_\omega T\wedge(dd^c\psi)^p+\lim_{s\to-\infty}
\int_{\Omega_\delta}u_sT\wedge(dd^c\psi)^p\cr
&\ge -\big(M+||\psi||_{L^\infty(\Omega)}M/\delta\big)
\int_\Omega T\wedge(dd^c\psi)^p.\cr}$$
The last integral is finite. This concludes the proof.\qed
\endproof

\claim{(4.2) Remark} \rm If $\Omega$ is smooth and strongly pseudoconvex, the above
proof shows in fact that
$$||uT||_{\ol\Omega}\le{C\over\delta}||u||_{L^\infty(
(\ol\Omega\ssm\Omega_\delta)\cap\Supp\,T)}||T||_{\ol\Omega}$$
when $L(u)\cap\Supp\,T\subset\Omega_\delta$. In fact, if $u$ is continuous
and if $\omega$ is chosen sufficiently small, the constant $M$ can be taken
arbitrarily close to $||u||_{L^\infty(
(\ol\Omega\ssm\Omega_\delta)\cap\Supp\,T)}$. Moreover, the maximum
principle implies 
$$||u_+||_{L^\infty(\ol\Omega\cap\Supp\,T)}=
||u_+||_{L^\infty(\partial\Omega\cap\Supp\,T)},$$
so we can achieve that $u<-\varepsilon$ on a neighborhood of $\ol\Omega
\cap\Supp\,T$ by subtracting $||u||_{L^\infty((\ol\Omega\ssm\Omega_\delta)
\cap\Supp\,T)}+2\varepsilon$ $[$Proof of maximum principle: if $u(z_0)>0$ at
$z_0\in\Omega\cap\Supp\,T$ and $u\le 0$ near $\partial\Omega\cap\Supp\,T$, 
then
$$\int_\Omega u_+T\wedge(dd^c\psi)^p=\int_\Omega\psi dd^cu_+\wedge T
\wedge(dd^c\psi)^{p-1}\le 0,$$
a contradiction$]$.\qed
\endclaim

\claim{(4.3) Corollary} Let $u_1,\ldots,u_q$ be plurisubharmonic functions on $X$
such that $X$ is covered by Stein open sets $\Omega$ with
$\partial\Omega\cap L(u_j)\cap\Supp\,T=\emptyset$.
We use again induction to define
$$dd^c u_1\wedge dd^cu_2\wedge\ldots\wedge dd^c u_q\wedge T=
dd^c(u_1dd^c u_2\ldots\wedge dd^c u_q\wedge T).$$
Then, if $u_1^k,\ldots,u_q^k$ are decreasing sequences of plurisubharmonic
functions converging pointwise to $u_1,\ldots,u_q$, $q\le p$,
properties~$(3.7\,{\rm a,b})$ hold.
\endclaim

\vbox{%
\input epsfiles/fig_3_2.tex
\vskip6mm
\centerline{{\bf III-2} Modified construction of $v_j^k$}}
\vskip5mm

\proof{Proof} Same proof as for Th.~3.7, with the following minor
modification: the max procedure $v_j^k:=\max\{u_j^k,A\psi\}$ is applied
only on a neighborhood $\omega$ of $\Supp\,T\cap(\ol\Omega\ssm\Omega_\delta)$
with $\delta>0$ small, and $u_j^k$ is left
unchanged near \hbox{$\Supp\,T\cap\ol\Omega_\delta$.} Observe that the
integration by part process requires the
functions $u_j^k$ and $\smash{u_j^{k,\varepsilon}}$ to be defined only 
near $\ol\Omega\cap\Supp\,T$.\qed
\endproof

\claim{(4.4) Proposition} Let $\Omega\compact X$ be a Stein open subset.
If $V$ is a plurisubharmonic function on $X$ and  $u_1,\ldots,u_q$,
$1\le q\le n-1$, are plurisubharmonic functions such that
$\partial\Omega\cap L(u_j)=\emptyset$,
then $Vdd^cu_1\wedge\ldots \wedge dd^cu_q$ has locally finite mass
in $\Omega$.
\endclaim

\proof{Proof} Same proof as for 3.11, when $\delta>0$ is taken so small that
$\Omega_\delta\supset L(u_j)$ for all $1\le j\le q$.\qed
\endproof

Finally, we show that Monge-Amp\`ere operators can also be defined
in the case of plurisubharmonic functions with non compact pole sets,
provided that the mutual intersections of the pole sets are of
sufficiently small Hausdorff dimension with respect to the dimension
$p$ of~$T$.

\claim{(4.5) Theorem} Let $u_1,\ldots,u_q$ be plurisubharmonic functions on~$X$.
The currents $u_1dd^cu_2\wedge\ldots\wedge dd^cu_q\wedge T$ and
$dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge T$ are well defined and have
locally finite mass in~$X$ as soon as $q\le p$ and
$$\cH_{2p-2m+1}\big(L(u_{j_1})\cap\ldots\cap L(u_{j_m})\cap\Supp\,T\big)=0$$
for all choices of indices $j_1<\ldots<j_m$ in $\{1,\ldots,q\}$.
\endclaim

The proof is an easy induction on $q$, thanks to the following improved
version of the Chern-Levine-Nirenberg inequalities.

\claim{(4.6) Proposition} Let $A_1,\ldots,A_q\subset X$ be closed sets
such that 
$$\cH_{2p-2m+1}\big(A_{j_1}\cap\ldots\cap A_{j_m}\cap\Supp\,T\big)=0$$
for all choices of $j_1<\ldots<j_m$ in $\{1,\ldots,q\}$. Then for all compact
sets $K$, $L$ of $X$ with $L\subset K^\circ$, there exist neighborhoods
$V_j$ of $K\cap A_j$ and a constant $C=C(K,L,A_j)$ such that the
conditions $u_j\le 0$ on $K$ and $L(u_j)\subset A_j$ imply
\medskip
\item{\rm a)} $||u_1dd^cu_2\wedge\ldots\wedge dd^cu_q\wedge T||_L\le
C||u_1||_{L^\infty(K\ssm V_1)}\ldots||u_q||_{L^\infty(K\ssm V_q)}||T||_K$
\medskip
\item{\rm b)} $||dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge T||_L\le
C||u_1||_{L^\infty(K\ssm V_1)}\ldots||u_q||_{L^\infty(K\ssm V_q)}||T||_K$.
\endclaim

\proof{Proof} We need only show that every point $x_0\in K^\circ$ has a neighborhood
$L$ such that a), b) hold. Hence it is enough to work in a coordinate open
set. We may thus assume that $X\subset\bC^n$ is open, and after a
regularization process $u_j=\lim u_j\star\rho_\varepsilon$ for $j=q$,
$q-1,\ldots,1$ in this order, that $u_1,\ldots,u_q$ are smooth.
We proceed by induction on $q$ in two steps:
\smallskip
\noindent{\it Step 1}.~~$({\rm b}_{q-1})\Longrightarrow({\rm b}_q),$\Fbreak
\noindent{\it Step 2}.~~$({\rm a}_{q-1})~{\rm and}~({\rm b}_q)
\Longrightarrow({\rm a}_q),$
\smallskip
\noindent where $({\rm b}_0)$ is the trivial statement $||T||_L\le||T||_K$
and $({\rm a}_0)$ is void. Observe that we have
$({\rm a}_q)\Longrightarrow({\rm a}_\ell)$
and $({\rm b}_q)\Longrightarrow({\rm b}_\ell)$ for $\ell\le q\le p$ by taking
$u_{\ell+1}(z)=\ldots=u_q(z)=|z|^2$. We need the following elementary fact.
\endproof

\claim{(4.7) Lemma} Let $F\subset\bC^n$ be a closed set such that $\cH_{2s+1}(F)=0$
for some integer $0\le s<n$. Then for almost all choices of unitary
coordinates $(z_1,\ldots,z_n)=(z',z'')$ with $z'=(z_1,\ldots,z_s)$,
$z''=(z_{s+1},\ldots,z_n)$ and almost all radii of balls
$B''=B(0,r'')\subset\bC^{n-s}$, the set $\{0\}\times\partial B''$
does not intersect~$F$.
\endclaim

\proof{Proof} The unitary group $U(n)$ has real dimension $n^2$. There is a proper
submersion
$$\Phi:U(n)\times\big(\bC^{n-s}\ssm\{0\}\big)\longrightarrow\bC^n\ssm\{0\},~~~~
(g,z'')\longmapsto g(0,z''),$$
whose fibers have real dimension $N=n^2-2s$. It follows that the inverse
image $\Phi^{-1}(F)$ has zero Hausdorff measure $\cH_{N+2s+1}=\cH_{n^2+1}$.
The set of pairs $(g,r'')\in U(n)\times\bR^\star_+$ such that
$g(\{0\}\times\partial B'')$ intersects $F$ is precisely the image of
$\Phi^{-1}(F)$ in $U(n)\times\bR^\star_+$ by the Lipschitz map
$(g,z'')\mapsto (g,|z''|)$. Hence this set has zero
$\cH_{n^2+1}$-measure.\qed
\endproof

\proof{Proof of step 1.} Take $x_0=0\in K^\circ$. Suppose first
$0\in A_1\cap\ldots\cap A_q$ and set $F=A_1\cap\ldots\cap A_q\cap\Supp\,T$.
Since $\cH_{2p-2q+1}(F)=0$, Lemma~4.7 implies that there are
coordinates $z'=(z_1,\ldots,z_s)$, $z''=(z_{s+1},\ldots,z_n)$ with $s=p-q$ and
a ball $\ol B''$ such that $F\cap\big(\{0\}\times\partial B''\big)=\emptyset$
and $\{0\}\times\ol B''\subset K^\circ$. By compactness of $K$, we can
find neighborhoods $W_j$ of $K\cap A_j$ and a ball $B'=B(0,r')\subset\bC^s$
such that $\ol B'\times\ol B''\subset K^\circ$ and
$$\ol W_1\cap\ldots\cap\ol W_q\cap\Supp\,T\cap\Big(\ol B'\times\big(
\ol B''\ssm(1-\delta)B''\big)\Big)=\emptyset\leqno(4.8)$$
for $\delta>0$ small. If $0\notin A_j$ for some~$j$, we choose instead
$W_j$ to be a small neighborhood of~$0$ such that $\ol W_j\subset
(\ol B'\times(1-\delta)B'')\ssm A_j\,$; property (4.8) is then automatically
satisfied. Let $\chi_j\ge 0$ be a function with compact support in $W_j$,
equal to $1$ near $K\cap A_j$ if $A_j\ni 0$ (resp.\ equal to $1$ near $0$
if $A_j\not\ni 0$) and let $\chi(z')\ge 0$ be a function equal to $1$
on $1/2\,B'$ with compact support in~$B'$. Then
$$\int_{B'\times B''}dd^c(\chi_1u_1)\wedge\ldots\wedge dd^c(\chi_qu_q)
\wedge T\wedge\chi(z')\,(dd^c|z'|^2)^s=0$$
because the integrand is $dd^c$ exact and has compact support 
in $B'\times B''$ thanks to~(4.8). If we expand all factors
$dd^c(\chi_ju_j)$, we find a term
$$\chi_1\ldots\chi_q\chi(z')dd^cu_1\wedge\ldots\wedge dd^c u_q\wedge T\ge 0$$
which coincides with $dd^cu_1\wedge\ldots\wedge dd^c u_q\wedge T$ on a
small neighborhood of $0$ where $\chi_j=\chi=1$. The other terms involve
$$d\chi_j\wedge d^cu_j+du_j\wedge d^c\chi_j+u_jdd^c\chi_j$$
for at least one index $j$. However $d\chi_j$ and $dd^c\chi_j$ vanish on
some neighborhood $V'_j$ of~$K\cap A_j$ and therefore $u_j$ is bounded on
$\smash{\ol B'\times\ol B''}\ssm V'_j$. We then apply the induction
hypothesis $({\rm b}_{q-1})$ to the current
$$\Theta=dd^cu_1\wedge\ldots\wedge\wh{dd^cu_j}\wedge
\ldots\wedge dd^cu_q\wedge T$$
and the usual Chern-Levine-Nirenberg inequality to the product of $\Theta$
with the mixed term $d\chi_j\wedge d^cu_j+du_j\wedge d^c\chi_j$.
Remark~3.5 can be applied because $\chi_j$ is smooth and is therefore a
difference $\chi^{(1)}_j-\chi^{(2)}_j$ of locally bounded plurisubharmonic
functions in $\bC^n$. Let $K'$ be a compact neighborhood of $\smash{
\ol B'\times\ol B''}$ with $K'\subset K^\circ$, and let $V_j$ be a 
neighborhood of $K\cap A_j$ with $\ol V_j\subset V'_j$. Then with
$L':=(\smash{\ol B'\times\ol B''})\ssm V'_j\subset(K'\ssm V_j)^\circ$
we obtain
$$\eqalign{
||(d\chi_j{\wedge}d^cu_j+du_j{\wedge}d^c&\chi_j)\wedge
\Theta||_{\ol B'\times \ol B''}
=||(d\chi_j{\wedge}d^cu_j+du_j{\wedge}d^c\chi_j)\wedge\Theta||_{L'}\cr
&\le C_1||u_j||_{L^\infty(K'\ssm V_j)}||\Theta||_{K'\ssm V_j},\cr
||\Theta||_{K'\ssm V_j}\le||\Theta||_{K'}
&\le C_2||u_1||_{L^\infty(K\ssm V_1)}\ldots\wh{||u_j||}\ldots
||u_q||_{L^\infty(K\ssm V_q)}||T||_K.\cr}$$
Now, we may slightly move the unitary basis in $\bC^n$ and get coordinate
systems $z^m=(z^m_1,\ldots,z^m_n)$ with the same properties as above, such that
the forms
$$(dd^c|z^{m\prime}|^2)^s={s!\over\pi^s}
\ii\,dz^m_1\wedge d\ol z^m_1\wedge\ldots\wedge\ii\,dz^m_s
\wedge d\ol z^m_s,~~~~1\le m\le N$$
define a basis of $\bigwedge^{s,s}(\bC^n)^\star$. It follows that all
measures
$$dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge T\wedge
\ii\,dz^m_1\wedge d\ol z^m_1\wedge\ldots\wedge\ii\,dz^m_s\wedge d\ol z^m_s$$
satisfy estimate $({\rm b}_q)$ on a small neighborhood $L$ of~$0$.
\endproof

\proof{Proof of Step 2.} We argue in a similar way with the integrals
$$\eqalign{
\int_{B'\times B''}&\chi_1u_1dd^c(\chi_2u_2)\wedge\ldots dd^c(\chi_qu_q)
\wedge T\wedge\chi(z')(dd^c|z'|^2)^s\wedge dd^c|z_{s+1}|^2\cr
&=\int_{B'\times B''}|z_{s+1}|^2dd^c(\chi_1u_1)\wedge
\ldots dd^c(\chi_qu_q)\wedge T\wedge\chi(z')(dd^c|z'|^2)^s.\cr}$$
We already know by $({\rm b}_q)$ and Remark~3.5 that all terms in the right
hand integral admit the desired bound. For $q=1$, this shows that
$({\rm b}_1)\Longrightarrow({\rm a}_1)$. Except for
$\chi_1\ldots\chi_q\chi(z')\,u_1dd^cu_2\wedge\ldots\wedge
dd^cu_q\wedge T$, all terms in the left hand integral involve derivatives
of~$\chi_j$. By construction, the support of these derivatives is disjoint
from~$A_j$, thus we only have to obtain a bound for
$$\int_Lu_1dd^cu_2\wedge\ldots\wedge dd^cu_q\wedge T\wedge\alpha$$
when $L=\ol B(x_0,r)$ is disjoint from $A_j$ for some $j\ge 2$,
say $L\cap A_2=\emptyset$, and $\alpha$ is a constant positive form of
type~$(p-q,p-q)$. Then $\ol B(x_0,r+\varepsilon)\subset K^\circ\ssm\ol V_2$
for some $\varepsilon>0$ and some neighborhood $V_2$ of~$K\cap A_2$. By the
max construction used e.g.\ in Prop.~4.1, we can replace $u_2$ by a
plurisubharmonic function $\wt u_2$ equal to $u_2$ in $L$ and
to $A(|z-x_0|^2-r^2)-M$ in $\ol B(x_0,r+\varepsilon)\ssm B(x_0,r+
\varepsilon/2)$, with $M=||u_2||_{L^\infty(K\ssm V_2)}$ and
$A=M/\varepsilon r$. Let $\chi\ge 0$ be a smooth function equal to $1$
on $B(x_0,r+\varepsilon/2)$ with support in $B(x_0,r)$. Then
$$\eqalign{
\int_{B(x_0,r+\varepsilon)}u_1&dd^c(\chi\wt u_2)\wedge dd^cu_3\wedge
\ldots\wedge dd^cu_q\wedge T\wedge\alpha\cr
&=\int_{B(x_0,r+\varepsilon)}\chi\wt u_2dd^cu_1\wedge dd^cu_3\wedge\ldots
\wedge dd^cu_q\wedge T\wedge\alpha\cr
&\le O(1)~||u_1||_{L^\infty(K\ssm V_1)}\ldots||u_q||_{L^\infty(K\ssm V_q)}
||T||_K\cr}$$
where the last estimate is obtained by the induction hypothesis
$({\rm b}_{q-1})$ applied to
$dd^cu_1\wedge dd^cu_3\wedge\ldots\wedge dd^cu_q\wedge T$. By construction
$$dd^c(\chi\wt u_2)=\chi\,dd^c\wt u_2+(\hbox{\rm smooth terms
involving $d\chi$})$$
coincides with $dd^cu_2$ in $L$, and $({\rm a}_{q-1})$ implies the
required estimate for the other terms in the left hand integral.\qed
\endproof

\claim{(4.9) Proposition} With the assumptions of Th.~$4.5$, the
analogue of the monotone convergence Theorem {\rm 3.7~(a,b)} holds.
\endclaim

\proof{Proof} By the arguments already used in the proof of Th.~3.7 (e.g.\ by
Lemma~3.9), it is enough to show that
$$\eqalign{
\int_{B'\times B''}&\chi_1\ldots\chi_q\,u_1\wedge dd^cu_2\wedge
\ldots\wedge dd^cu_q\wedge T\wedge\alpha\cr
&\le\liminf_{k\to+\infty}\int_{B'\times B''}\chi_1\ldots\chi_q\,
u^k_1dd^cu^k_2\wedge\ldots\wedge dd^cu^k_q\wedge T\wedge\alpha\cr}$$
where $\alpha=\chi(z')(dd^c|z'|^2)^s$ is closed.
Here the functions $\chi_j$, $\chi$ are chosen as in the proof of Step~1
in~4.7, especially their product has compact support in $B'\times B''$
and $\chi_j=\chi=1$ in a neighborhood of the given point~$x_0$. We argue
by induction on $q$ and also on the number $m$ of functions $(u_j)_{j\ge 1}$
which are unbounded near~$x_0$. If $u_j$ is bounded near $x_0$, we take $W_j''
\compact W'_j\compact W_j$ to be small balls of center $x_0$ on which $u_j$
is bounded and we modify the sequence $u_j^k$ on the corona $W_j\ssm W''_j$
so as to make it constant and equal to a smooth function $A|z-x_0|^2+B$ on
the smaller corona~$W_j\ssm W'_j$. In that case, we take $\chi_j$
equal to $1$ near~$\smash{\ol W'_j}$ and $\Supp\,\chi_j\subset W_j$.
For every $\ell=1,\ldots,q$, we are going to check that
$$\eqalign{
\liminf_{k\to+\infty}\int_{B'\times B''}
\chi_1u^k_1&dd^c(\chi_2u^k_2)\wedge\ldots\cr
dd^c(\chi_{\ell-1}u^k_{\ell-1})\wedge{}&dd^c(\chi_\ell u_\ell)\wedge
dd^c(\chi_{\ell+1}u_{\ell+1})\ldots dd^c(\chi_qu_q)\wedge T\wedge\alpha\cr
{}\le\liminf_{k\to+\infty}\int_{B'\times B''}
\chi_1u^k_1&dd^c(\chi_2u^k_2)\wedge\ldots\cr
dd^c(\chi_{\ell-1}u^k_{\ell-1})\wedge{}&dd^c(\chi_\ell u^k_\ell)\wedge
dd^c(\chi_{\ell+1}u_{\ell+1})\ldots dd^c(\chi_qu_q)\wedge T\wedge\alpha.\cr}$$
In order to do this, we integrate by parts $\chi_1u^k_1dd^c(\chi_\ell
u_\ell)$ into $\chi_\ell u_\ell dd^c(\chi_1u^k_1)$ for $\ell\ge 2$,
and we use the inequality $u_\ell\le u^k_\ell$. Of course, the derivatives
$d\chi_j$, $d^c\chi_j$, $dd^c\chi_j$ produce terms which are no longer
positive and we have to take care of these. However, $\Supp\,d\chi_j$ is
disjoint from the unbounded locus of $u_j$ when $u_j$ is
unbounded, and contained in $W_j\ssm\smash{\ol W'_j}$ when $u_j$ is bounded.
The number $m$ of unbounded functions is therefore replaced by $m-1$ in the
first case, whereas in the second case $u^k_j=u_j$ is constant and smooth
on $\Supp\,d\chi_j$, so $q$ can be replaced by $q-1$. By induction on $q+m$
(and thanks to the polarization technique~3.5), the limit of the terms
involving derivatives of $\chi_j$ is equal on both sides to the
corresponding terms obtained by suppressing all indices~$k$. Hence
these terms do not give any contribution in the inequalities.\qed
\endproof

We finally quote the following simple consequences of Th.~4.5 when 
$T$ is arbitrary and $q=1$, resp.\ when $T=1$ has bidegree $(0,0)$ and
$q$ is arbitrary.

\claim{(4.10) Corollary} Let $T$ be a closed positive current of bidimension $(p,p)$
and let $u$ be a plurisubharmonic function on $X$ such that $L(u)\cap\Supp\,T$
is contained in an analytic set of dimension at most $p-1$. Then
$uT$ and $dd^cu\wedge T$ are well defined and have locally finite mass
in~$X$.\qed
\endclaim

\claim{(4.11) Corollary} Let $u_1,\ldots,u_q$ be plurisubharmonic functions on $X$
such that $L(u_j)$ is contained in an analytic set $A_j\subset X$ for 
every~$j$. Then $dd^cu_1\wedge\ldots\wedge dd^cu_q$ is well defined as
soon as $A_{j_1}\cap\ldots\cap A_{j_m}$ has codimension at least $m$ for
all choices of indices $j_1<\ldots<j_m$ in $\{1,\ldots,q\}$.\qed
\endclaim

In the particular case when $u_j=\log|f_j|$ for some non zero
holomorphic function $f_j$ on $X$, we see that the intersection
product of the associated zero divisors $[Z_j]=dd^cu_j$ is well
defined as soon as the supports $|Z_j|$ satisfy
$\codim|Z_{j_1}|\cap\ldots\cap |Z_{j_m}|=m$ for every~$m$. Similarly,
when $T=[A]$ is an analytic $p$-cycle, Cor.~4.10 shows that
$[Z]\wedge[A]$ is well defined for every divisor $Z$ such that
$\dim|Z|\cap|A|=p-1$. These observations easily imply the following

\claim{(4.12) Proposition} Suppose that the divisors $Z_j$ satisfy the
above codimension condition and let $(C_k)_{k\ge 1}$ be the irreducible
components of the point set intersection $|Z_1|\cap\ldots\cap|Z_q|$. Then
there exist integers $m_k>0$ such that
$$[Z_1]\wedge\ldots\wedge[Z_q]=\sum m_k[C_k].$$
The integer $m_k$ is called the multiplicity of intersection of $Z_1,\ldots,Z_q$
along the component~$C_k$.
\endclaim

\proof{Proof} The wedge product has bidegree $(q,q)$ and support in
$C=\bigcup C_k$ where $\codim C=q$, so
it must be a sum as above with $m_k\in\bR_+$. We check by induction
on $q$ that $m_k$ is a positive integer. If we denote by $A$ some
irreducible component of $|Z_1|\cap\ldots\cap |Z_{q-1}|$, we need
only check that $[A]\wedge[Z_q]$ is an integral analytic cycle of
codimension $q$ with positive coefficients on each component $C_k$ of
the intersection. However $[A]\wedge[Z_q]=dd^c(\log|f_q|~[A])$. First
suppose that no component of $A\cap f_q^{-1}(0)$ is contained in the
singular part $A_\sing$. Then the Lelong-Poincar\'e equation applied
on $A_\reg$ shows that $dd^c(\log|f_q|~[A])=\sum m_k[C_k]$ on $X\ssm
A_\sing$, where $m_k$ is the vanishing order of $f_q$ along $C_k$ in
$A_\reg$. Since $C\cap A_\sing$ has codimension $q+1$ at least, the
equality must hold on $X$. In general, we replace $f_q$ by
$f_q-\varepsilon$ so that the divisor of $f_q-\varepsilon$ has no
component contained in $A_\sing$. Then
$dd^c(\log|f_q-\varepsilon|~[A])$ is an integral codimension $q$ cycle
with positive multiplicities on each component of $A\cap
f_q^{-1}(\varepsilon)$ and we conclude by letting $\varepsilon$ tend
to zero.\qed
\endproof

\section{5}{Generalized Lelong Numbers}
The concepts we are going to study mostly concern the behaviour of
currents or plurisubharmonic functions in a neighborhood of a point
at which we have for instance a logarithmic pole. Since the interesting
applications are local, we assume from now on (unless otherwise stated)
that $X$ is a Stein manifold, i.e.\ that $X$ has a strictly plurisubharmonic
exhaustion function. Let $\varphi:X\longrightarrow[-\infty,+\infty[$ 
be a continuous plurisubharmonic function (in general $\varphi$ may have
$-\infty$ poles, our continuity assumption means that $e^\varphi$ is
continuous). The sets
$$\leqalignno{
S(r)&=\{x\in X\,;\,\varphi(x)=r\},&(5.1)\cr
B(r)&=\{x\in X\,;\,\varphi(x)<r\},&(5.1')\cr
\ol B(r)&=\{x\in X\,;\,\varphi(x)\le r\}&(5.1'')\cr}$$
will be called {\it pseudo-spheres} and {\it pseudo-balls}
associated with $\varphi$. Note that $\ol B(r)$ is not necessarily equal
to the closure of $B(r)$, but this is often true in concrete situations.
The most simple example we have in mind is the case of the
function $\varphi(z)=\log|z-a|$ on
an open subset $X\subset\bC^n\,$; in this case $B(r)$ is the euclidean
ball of center $a$ and radius $e^r\,$; moreover, the forms
$${1\over 2}dd^ce^{2\varphi}={\ii\over 2\pi}d'd''|z|^2,~~~~
dd^c\varphi={\ii\over\pi}d'd''\log|z-a|\leqno(5.2)$$
can be interpreted respectively as the flat hermitian metric on $\bC^n$
and as the pull-back over $\bC^n$ of the Fubini-Study
metric of $\bP^{n-1}$, translated by $a$.

\claim{(5.3) Definition} We say that $\varphi$ is semi-exhaustive
if there exists a real number $R$ such that $B(R)\compact X$. 
Similarly, $\varphi$ is said to be
semi-exhaustive on a closed subset $A\subset X$ if there exists
$R$ such that $A\cap B(R)\compact X$.
\endclaim

We are interested especially in the set of poles $S(-\infty)=
\{\varphi=-\infty\}$ and in the behaviour of $\varphi$ near $S(-\infty)$.
Let $T$ be a closed positive current of bidimension $(p,p)$ on $X$.
Assume that $\varphi$ is semi-exhaustive on $\Supp\,T$ and that
$B(R)\cap\Supp\,T\compact X$.
Then $P=S(-\infty)\cap{\rm Supp T}$ is compact and the results
of \S 2 show that the measure $T\wedge(dd^c\varphi)^p$ 
is well defined. Following \cite{Demailly 1982b, 1987a}, we introduce:

\claim{(5.4) Definition} If $\varphi$ is semi-exhaustive on $\Supp\,T$ and if
$R$ is such that $B(R)\cap\Supp\,T\compact X$, we set for all
$r\in{}]-\infty,R[$
$$\eqalign{
\nu(T,\varphi,r)&=\int_{B(r)}T\wedge(dd^c\varphi)^p,\cr
\nu(T,\varphi)&=\int_{S(-\infty)}T\wedge(dd^c\varphi)^p=\lim_{r\to-\infty}
\nu(T,\varphi,r).\cr}$$
The number $\nu(T,\varphi)$ will be called the (generalized) Lelong
number of $T$ with respect to the weight $\varphi$.
\endclaim

If we had not required $T\wedge(dd^c\varphi)^p$ to be
defined pointwise on $\varphi^{-1}(-\infty)$, the assumption
that $X$ is Stein could have been dropped: in fact, the integral over
$B(r)$ always makes sense if we define
$$\nu(T,\varphi,r)=\int_{B(r)}T\wedge\big(dd^c\max\{\varphi,s\}\big)^p~~~~
\hbox{\rm with}~~s<r.$$
Stokes' formula shows that the right hand integral is
actually independent of~$s$. The example given after (4.1) shows however
that $T\wedge(dd^c\varphi)^p$ need not exist on $\varphi^{-1}(-\infty)$
if $\varphi^{-1}(-\infty)$ contains an exceptional compact analytic
subset. We leave the reader consider by himself this more general situation
and extend our statements by the $\max\{\varphi,s\}$ technique.
Observe that $r\longmapsto\nu(T,\varphi,r)$ is always an increasing
function of $r$. Before giving examples, we need a formula.

\claim{(5.5) Formula} For any convex increasing function 
$\chi:\bR\longrightarrow\bR$ we have
$$\int_{B(r)}T\wedge(dd^c\chi\circ\varphi)^p=
\chi'(r-0)^p\,\nu(T,\varphi,r)$$
where $\chi'(r-0)$ denotes the left derivative of $\chi$ at $r$.
\endclaim

\proof{Proof} Let $\chi_\varepsilon$ be the convex function equal to
$\chi$ on $[r-\varepsilon,+\infty[$ and to
a linear function of slope $\chi'(r-\varepsilon-0)$
on $]-\infty,r-\varepsilon]$. We get
$dd^c(\chi_\varepsilon\circ\varphi)=\chi'(r-\varepsilon-0)dd^c\varphi$
on $B(r-\varepsilon)$ and Stokes' theorem implies
$$\eqalign{
\int_{B(r)}T\wedge(dd^c\chi\circ\varphi)^p
&=\int_{B(r)}T\wedge(dd^c\chi_\varepsilon\circ\varphi)^p\cr
&\ge\int_{B(r-\varepsilon)}T\wedge(dd^c\chi_\varepsilon\circ\varphi)^p\cr
&=\chi'(r-\varepsilon-0)^p\nu(T,\varphi,r-\varepsilon).\cr}$$
Similarly, taking $\wt\chi_\varepsilon$ equal to $\chi$ on 
$]-\infty,r-\varepsilon]$ and linear on $[r-\varepsilon,r]$, we obtain
$$\int_{B(r-\varepsilon)}T\wedge(dd^c\chi\circ\varphi)^p
\le\int_{B(r)}T\wedge(dd^c\wt\chi_\varepsilon\circ\varphi)^p
=\chi'(r-\varepsilon-0)^p\nu(T,\varphi,r).$$
The expected formula follows when $\varepsilon$ tends to $0$.\qed
\endproof

We get in particular
$\int_{B(r)}T\wedge(dd^ce^{2\varphi})^p=(2e^{2r})^p\nu(T,\varphi,r)$,
whence the formula
$$\nu(T,\varphi,r)=e^{-2pr}\int_{B(r)}T\wedge
\Big({1\over 2}dd^ce^{2\varphi}\Big)^p.\leqno(5.6)$$

Now, assume that $X$ is an open subset of $\bC^n$ and that 
$\varphi(z)=\log|z-a|$ for some $a\in X$. Formula (5.6) gives
$$\nu(T,\varphi,\log r)=r^{-2p}\int_{|z-a|<r}T\wedge\Big({\ii\over 2\pi}
d'd''|z|^2\Big)^p.$$
The positive measure $\sigma_T={1\over p!}T\wedge({\ii\over 2}d'd''|z|^2)^p
=2^{-p}\sum T_{I,I}\,.\,\ii^n dz_1\wedge\ldots\wedge d\ol z_n$ 
is called the {\it trace measure} of $T$. We get
$$\nu(T,\varphi,\log r)={\sigma_T\big(B(a,r)\big)\over
\pi^p r^{2p}/p!}\leqno(5.7)$$
and $\nu(T,\varphi)$ is the limit of this ratio as $r\to 0$. This
limit is called the ({\it ordinary}) {\it Lelong number}
of $T$ at point~$a$ and is denoted~$\nu(T,a)$.
This was precisely the original definition of Lelong, see \cite{Lelong 1968}.
Let us mention a simple but important consequence.

\claim{(5.8) Consequence} The ratio $\sigma_T\big(B(a,r)\big)/r^{2p}$
is an increasing function of~$r$. Moreover, for every compact
subset $K\subset X$ and every $r_0<d(K,\partial X)$ we have
$$\sigma_T\big(B(a,r)\big)\le Cr^{2p}~~~~\hbox{\rm for}~~a\in K~
\hbox{\rm and}~r\le r_0,$$
where $C=\sigma_T\big(K+\ol B(0,r_0)\big)/r_0^{2p}$.
\endclaim

All these results are particularly interesting when $T=[A]$ is the 
current of integration
over an analytic subset $A\subset X$ of pure dimension $p$. Then
$\sigma_T\big(B(a,r)\big)$ is the euclidean area of $A\cap B(a,r)$,
while $\pi^pr^{2p}/p!$ is the area of a ball of radius $r$ in a
$p$-dimensional subspace of $\bC^n$. Thus $\nu(T,\varphi,\log r)$ is
the ratio of these areas and the Lelong number $\nu(T,a)$
is the limit ratio.

\claim{(5.9) Remark} \rm It is immediate to check that 
$$\nu([A],x)=\cases{0&for~ $x\notin A$,\cr
                    1&when $x\in A$ is a regular point.\cr}$$
We will see later that $\nu([A],x)$ is always an integer (Thie's
theorem~8.7).
\endclaim

\claim{(5.10) Remark} \rm When $X=\bC^n$, $\varphi(z)=\log|z-a|$ and 
$A=X$ (i.e.\ $T=1$), we obtain in particular $\int_{B(a,r)}
(dd^c\log|z-a|)^n=1$ for all $r$. This implies 
$$(dd^c\log|z-a|)^n=\delta_a.$$
This fundamental formula can be viewed as a higher dimensional analogue
of the usual formula $\Delta\log|z-a|=2\pi\delta_a$ in $\bC$.\qed
\endclaim

We next prove a result which shows in particular
that the Lelong numbers of a closed positive current are zero except
on a very small set.

\claim{(5.11) Proposition} If $T$ is a closed positive current of
bidimension $(p,p)$, then for each $c>0$ the set
$E_c=\{x\in X\,;\,\nu(T,x)\ge c\}$ is a closed set of locally finite
$\cH_{2p}$ Hausdorff measure in $X$.
\endclaim

\proof{Proof} By (5.7), we infer
$\nu(T,a)=\lim_{r\to 0}\sigma_T\big(\ol B(a,r)\big)p!/\pi^pr^{2p}$.
The function $a\mapsto\sigma_T\big(\ol B(a,r)\big)$ is clearly
upper semicontinuous. Hence the decreasing limit $\nu(T,a)$ as
$r$ decreases to $0$ is also upper semicontinuous in~$a$. This implies
that $E_c$ is closed. Now, let $K$ be a compact subset in $X$ and
let $\{a_j\}_{1\le j\le N}$, $N=N(\varepsilon)$, be a maximal collection
of points in $E_c\cap K$ such that $|a_j-a_k|\ge 2\varepsilon$ for $j\ne k$.
The balls $B(a_j,2\varepsilon)$ cover $E_c\cap K$,
whereas the balls $B(a_j,\varepsilon)$ are disjoint.
If $K_{c,\varepsilon}$ is the set of points which are at distance
$\le\varepsilon$ of $E_c\cap K$, we get
$$\sigma_T(K_{c,\varepsilon})\ge\sum\sigma_T\big(B(a_j,\varepsilon)\big)
\ge N(\varepsilon)\,c\pi^p\varepsilon^{2p}/p!,$$
since $\nu(T,a_j)\ge c$. By the definition of Hausdorff measure, we infer
$$\eqalignno{
\cH_{2p}(E_c\cap K)&\le\liminf_{\varepsilon\to 0}
\sum\big(\hbox{\rm diam}\,B(a_j,2\varepsilon)\big)^{2p}\cr
&\le\liminf_{\varepsilon\to 0}N(\varepsilon)(4\varepsilon)^{2p}
\le{p!4^{2p}\over c\pi^p}\sigma_T(E_c\cap K).&\square\cr}$$
\endproof

Finally, we conclude this section by proving two simple
semi-continuity results for Lelong numbers.

\claim{(5.12) Proposition} Let $T_k$ be a sequence of closed positive
currents of bidimension $(p,p)$ converging weakly to a limit~$T$.
Suppose that there is a closed set $A$ such that $\Supp\,T_k\subset A$
for all~$k$ and such that $\varphi$ is semi-exhaustive on $A$
with $A\cap B(R)\compact X$. Then for all $r<R$ we have
$$\eqalign{
\int_{B(r)}T\wedge(dd^c\varphi)^p
&\le\liminf_{k\to+\infty}\int_{B(r)}T_k\wedge(dd^c\varphi)^p\cr
&\le\limsup_{k\to+\infty}\int_{\ol B(r)}T_k\wedge(dd^c\varphi)^p
\le\int_{\ol B(r)}T\wedge(dd^c\varphi)^p.\cr}$$
When $r$ tends to $-\infty$, we find in particular
$$\limsup_{k\to+\infty}\nu(T_k,\varphi)\le\nu(T,\varphi).$$
\endclaim

\proof{Proof} Let us prove for instance the third inequality. Let $\varphi_\ell$
be a sequence of smooth plurisubharmonic approximations of $\varphi$ with
$\varphi\le\varphi_\ell<\varphi+1/\ell$ on $\{r-\varepsilon\le\varphi\le
r+\varepsilon\}$. We set
$$\psi_\ell=\cases{
\varphi&on $\ol B(r)$,\cr
\max\{\varphi,(1+\varepsilon)(\varphi_\ell-1/\ell)-r\varepsilon\}
&on $X\ssm B(r)$.\cr}$$
This definition is coherent since $\psi_\ell=\varphi$ near $S(r)$, and we have
$$\psi_\ell=(1+\varepsilon)(\varphi_\ell-1/\ell)-r\varepsilon~~~~
\hbox{\rm near}~~S(r+\varepsilon/2)$$
as soon as $\ell$ is large enough, i.e.\ 
$(1+\varepsilon)/\ell\le\varepsilon^2/2$. Let $\chi_\varepsilon$
be a cut-off function equal to $1$ in $B(r+\varepsilon/2)$ with support
in $B(r+\varepsilon)$. Then
$$\eqalign{
\int_{\ol B(r)}T_k\wedge(dd^c\varphi)^p
&\le\int_{B(r+\varepsilon/2)}T_k\wedge(dd^c\psi_\ell)^p\cr
&=(1+\varepsilon)^p\int_{B(r+\varepsilon/2)}T_k\wedge(dd^c\varphi_\ell)^p\cr
&\le(1+\varepsilon)^p\int_{B(r+\varepsilon)}\chi_\varepsilon
T_k\wedge(dd^c\varphi_\ell)^p.\cr}$$
As $\chi_\varepsilon(dd^c\varphi_\ell)^p$ is smooth with compact support and
as $T_k$ converges weakly to~$T$, we infer
$$\limsup_{k\to+\infty}\int_{\ol B(r)}T_k\wedge(dd^c\varphi)^p
\le(1+\varepsilon)^p\int_{B(r+\varepsilon)}\chi_\varepsilon
T\wedge(dd^c\varphi_\ell)^p.$$
We then let $\ell$ tend to $+\infty$ and $\varepsilon$ tend to $0$ to get
the desired inequality. The first inequality is obtained in a similar way,
we define $\psi_\ell$ so that $\psi_\ell=\varphi$ on $X\ssm B(r)$ and
$\psi_\ell=\max\{(1-\varepsilon)(\varphi_\ell-1/\ell)+r\varepsilon\}$ on
$\ol B(r)$, and we take $\chi_\varepsilon=1$ on $B(r-\varepsilon)$
with $\Supp\,\chi_\varepsilon\subset B(r-\varepsilon/2)$. Then
for $\ell$ large
$$\eqalignno{
\int_{B(r)}T_k\wedge(dd^c\varphi)^p
&\ge\int_{B(r-\varepsilon/2)}T_k\wedge(dd^c\psi_\ell)^p\cr
&\ge(1-\varepsilon)^p\int_{B(r-\varepsilon/2)}\chi_\varepsilon
T_k\wedge(dd^c\varphi_\ell)^p.&\square\cr}$$
\endproof

\claim{(5.13) Proposition} Let $\varphi_k$ be a $($non necessarily
monotone$)$ sequence of continuous plurisubharmonic functions such that
$e^{\varphi_k}$ converges uniformly to $e^\varphi$ on every compact
subset of $X$. Suppose that $\{\varphi<R\}\cap\Supp\,T\compact X$.
Then for $r<R$ we have
$$\limsup_{k\to+\infty}\int_{\{\varphi_k\le r\}\cap\{\varphi<R\}}
T\wedge(dd^c\varphi_k)^p\le
\int_{\{\varphi\le r\}}T\wedge(dd^c\varphi)^p.$$
In particular $\limsup_{k\to+\infty}\nu(T,\varphi_k)\le\nu(T,\varphi)$.
\endclaim

When we take $\varphi_k(z)=\log|z-a_k|$ with $a_k\to a$,
Prop.~5.13 implies the upper semicontinuity of $a\mapsto\nu(T,a)$ which
was already noticed in the proof of Prop.~5.11.

\proof{Proof} Our assumption is equivalent to saying that $\max\{\varphi_k,t\}$
converges locally uniformly to $\max\{\varphi,t\}$ for every~$t$.
Then Cor.~3.6 shows that $T\wedge(dd^c\max\{\varphi_k,t\})^p$
converges weakly to $T\wedge(dd^c\max\{\varphi,t\})^p$. If $\chi_\varepsilon$
is a cut-off function equal to $1$ on $\{\varphi\le r+\varepsilon/2\}$
with support in $\{\varphi<r+\varepsilon\}$, we get
$$\lim_{k\to+\infty}\int_X\chi_\varepsilon T\wedge(dd^c\max\{\varphi_k,t\})^p=
\int_X\chi_\varepsilon T\wedge(dd^c\max\{\varphi,t\})^p.$$
For $k$ large, we have $\{\varphi_k\le r\}\cap\{\varphi<R\}\subset
\{\varphi<r+\varepsilon/2\}$, thus when $\varepsilon$ tends to $0$ we infer
$$\limsup_{k\to+\infty}\int_{\{\varphi_k\le r\}\cap\{\varphi<R\}}
T\wedge(dd^c\max\{\varphi_k,t\})^p\le
\int_{\{\varphi\le r\}}T\wedge(dd^c\max\{\varphi,t\})^p.$$
When we choose $t<r$, this is equivalent to the first inequality in
statement (5.13).\qed
\endproof

\section{6}{The Jensen-Lelong Formula}
We assume in this section that $X$ is Stein, that $\varphi$ is
{\it semi-exhaustive} on $X$ and that $B(R)\compact X$. We set
for simplicity $\varphi_{\gge r}=\max\{\varphi,r\}$.
For every $r\in{}]-\infty,R[$, the measures
$dd^c(\varphi_{\gge r})^n$ are well defined.  By Cor.~3.6, the map 
$r\longmapsto(dd^c\varphi_{\gge r})^n$ is continuous on $]-\infty,R[$
with respect to the weak topology. As
$(dd^c\varphi_{\gge r})^n=(dd^c\varphi)^n$ on $X\setminus\ol B(r)$
and as $\varphi_{\gge r}\equiv r$, $(dd^c\varphi_{\gge r})^n=0$
on~$B(r)$, the left continuity implies 
$(dd^c\varphi_{\gge r})^n\ge\bOne_{X\setminus B(r)}(dd^c\varphi)^n$. 
Here $\bOne_A$ denotes the characteristic function of any subset
$A\subset X$. According to the definition introduced in 
\cite{Demailly 1985a},
the collection of {\it Monge-Amp\`ere measures} associated with $\varphi$ is 
the family of positive measures $\mu_r$ such that
$$\mu_r=(dd^c\varphi_{\gge r})^n-\bOne_{X\setminus B(r)}(dd^c\varphi)^n,
~~~~r\in{}]-\infty,R[.\leqno(6.1)$$ 
The measure $\mu_r$ is supported on $S(r)$ and $r\longmapsto\mu_r$ is 
weakly continuous on the left by the bounded convergence theorem. 
Stokes' formula shows
that $\int_{B(s)}(dd^c\varphi_{\gge r})^n-(dd^c\varphi)^n=0$ for $s>r$,
hence the total mass $\mu_r(S(r))=\mu_r(B(s))$ is equal
to the difference between the masses of $(dd^c\varphi)^n$ and 
$\bOne_{X\setminus B(r)}(dd^c\varphi)^n$ over $B(s)$, i.e.\
$$\mu_r\big(S(r)\big)=\int_{B(r)}(dd^c\varphi)^n.\leqno(6.2)$$

\claim{(6.3) Example} \rm When $(dd^c\varphi)^n=0$ on
$X\setminus\varphi^{-1}(-\infty)$, formula (6.1) can be simplified into
$\mu_r=(dd^c\varphi_{\gge r})^n$. This is so for $\varphi(z)=\log|z|$.
In this case, the invariance of $\varphi$
under unitary transformations implies that $\mu_r$ is also invariant.
As the total mass of $\mu_r$ is equal to $1$ by 5.10 and (6.2), we see
that $\mu_r$ is the invariant measure of mass $1$ on the euclidean sphere
of radius~$e^r$.
\endclaim

\claim{(6.4) Proposition} Assume that $\varphi$ is smooth near $S(r)$ and
that $d\varphi\ne 0$ on $S(r)$, i.e.\ $r$ is a non critical value. Then
$S(r)=\partial B(r)$ is a smooth oriented real hypersurface and
the measure $\mu_r$ is given by the $(2n-1)$-volume form
$(dd^c\varphi)^{n-1}\wedge d^c\varphi_{\restriction S(r)}$.
\endclaim

\proof{Proof} Write $\max\{t,r\}=\lim_{k\to+\infty}\chi_k(t)$ where $\chi$ is
a decreasing sequence of smooth convex functions with
$\chi_k(t)=r$ for $t\le r-1/k$, $\chi_k(t)=t$ for $t\ge r+1/k$.
Theorem 3.6 shows that $(dd^c\chi_k\circ\varphi)^n$ converges
weakly to $(dd^c\varphi_{\gge r})^n$.
Let $h$ be a smooth function $h$ with compact support near $S(r)$.
Let us apply Stokes' theorem with $S(r)$ considered as the boundary
of $X\setminus B(r)\,$:
$$\leqalignno{
\int_X h(dd^c\varphi_{\gge r})^n&=\lim_{k\to+\infty}
\int_X h(dd^c\chi_k\circ\varphi)^n\cr
&=\lim_{k\to+\infty}\int_X -dh\wedge(dd^c\chi_k\circ\varphi)^{n-1}
\wedge d^c(\chi_k\circ\varphi)\cr
&=\lim_{k\to+\infty}\int_X -\chi'_k(\varphi)^n\,dh\wedge(dd^c\varphi)^{n-1}
\wedge d^c\varphi\cr
&=\int_{X\setminus B(r)}-dh\wedge(dd^c\varphi)^{n-1}\wedge d^c\varphi\cr
&=\int_{S(r)}h\,(dd^c\varphi)^{n-1}\wedge d^c\varphi+
\int_{X\setminus B(r)}h\,(dd^c\varphi)^{n-1}\wedge dd^c\varphi.\cr}$$
Near $S(r)$ we thus have an equality of measures
$$(dd^c\varphi_{\gge r})^n=(dd^c\varphi)^{n-1}\wedge d^c
\varphi_{\restriction S(r)}+\bOne_{X\setminus B(r)}(dd^c\varphi)^n.
\eqno{\square}$$
\endproof

\claim{(6.5) Jensen-Lelong formula} Let $V$ be any plurisubharmonic function 
on~$X$. Then $V$ is $\mu_r$-integrable for every $r\in{}]-\infty,R[$ and
$$\mu_r(V)-\int_{B(r)}V(dd^c\varphi)^n=
\int_{-\infty}^r\nu(dd^cV,\varphi,t)\,dt.$$
\endclaim

\proof{Proof} Proposition 3.11 shows that $V$ is integrable with respect to
the measure $(dd^c\varphi_{\gge r})^n$, hence $V$ is $\mu_r$-integrable.
By definition 
$$\nu(dd^cV,\varphi,t)=\int_{\varphi(z)<t}dd^cV\wedge(dd^c\varphi)^{n-1}$$
and the Fubini theorem gives
$$\leqalignno{
\int_{-\infty}^r\nu(dd^cV,\varphi,t)\,dt
&=\int\!\!\!\int_{\varphi(z)<t<r}dd^cV(z)\wedge(dd^c\varphi(z))^{n-1}\,dt\cr
&=\int_{B(r)}(r-\varphi)dd^cV\wedge(dd^c\varphi)^{n-1}.&(6.6)\cr}$$
We first show that Formula~6.5 is true when $\varphi$ and $V$ are
smooth. As both members of the formula are left continuous with
respect to $r$ and as almost all values of $\varphi$ are non critical
by Sard's theorem, we may assume $r$ non critical. Formula~3.1 applied
with $f=(r-\varphi)(dd^c\varphi)^{n-1}$ and $g=V$ shows 
that integral $(6.6)$ is equal to
$$\int_{S(r)}V(dd^c\varphi)^{n-1}\wedge d^c\varphi-\int_{B(r)}
V\,(dd^c\varphi)^n=\mu_r(V)-\int_{B(r)}V\,(dd^c\varphi)^n.$$
Formula~6.5 is thus proved when $\varphi$ and $V$ are smooth.
If $V$ is smooth and $\varphi$ merely continuous and finite,
one can write $\varphi=\lim\varphi_k$ where $\varphi_k$ is a 
decreasing sequence of smooth plurisubharmonic functions 
(because $X$ is Stein). Then $dd^cV\wedge(dd^c\varphi_k)^{n-1}$
converges weakly to $dd^cV\wedge(dd^c\varphi)^{n-1}$ and (6.6)
converges, since $\bOne_{B(r)}(r-\varphi)$ is continuous with compact
support on $X$. The left hand side of Formula~6.5 also
converges because the definition of $\mu_r$ implies
$$\mu_{k,r}(V)-\int_{\varphi_k<r}V(dd^c\varphi_k)^n=
\int_X V\big((dd^c\varphi_{k,\gge r})^n-(dd^c\varphi_k)^n\big)$$
and we can apply again weak convergence on a neighborhood of
$\ol B(r)$. If $\varphi$ takes $-\infty$ values, replace
$\varphi$ by $\varphi_{\gge-k}$ where $k\to+\infty$. Then $\mu_r(V)$
is unchanged, $\int_{B(r)}V(dd^c\varphi_{\gge -k})^n$ converges
to $\int_{B(r)}V(dd^c\varphi)^n$ and the right hand side of Formula~6.5
is replaced by $\int_{-k}^r\nu(dd^cV,\varphi,t)\,dt$.
Finally, for $V$ arbitrary, write $V=\lim\downarrow V_k$
with a sequence of smooth functions $V_k$.
Then $dd^cV_k\wedge(dd^c\varphi)^{n-1}$ converges weakly to 
$dd^cV\wedge(dd^c\varphi)^{n-1}$ by Prop.~4.4, so the integral
(6.6) converges to the expected limit and the same is true for the
left hand side of 6.5 by the monotone convergence theorem.\qed
\endproof

For $r<r_0<R$, the Jensen-Lelong formula implies
$$\mu_r(V)-\mu_{r_0}(V)+\int_{B(r_0)\setminus B(r)}V(dd^c\varphi)^n=
\int_{r_0}^r\nu(dd^cV,\varphi,t)\,dt.\leqno(6.7)$$

\claim{(6.8) Corollary} Assume that $(dd^c\varphi)^n=0$ on $X\setminus
S(-\infty)$. Then \hbox{$r\mapsto\mu_r(V)$} is a convex increasing
function of $r$ and the lelong number $\nu(dd^cV,\varphi)$ is given by
$$\nu(dd^cV,\varphi)=\lim_{r\to-\infty}{\mu_r(V)\over r}.$$
\endclaim

\proof{Proof} By (6.7) we have
$$\mu_r(V)=\mu_{r_0}(V)+\int_{r_0}^r\nu(dd^cV,\varphi,t)\,dt.$$
As $\nu(dd^cV,\varphi,t)$ is increasing and nonnegative, it follows
that $r\longmapsto\mu_r(V)$ is convex and increasing. The formula
for $\nu(dd^cV,\varphi)=\lim_{t\to-\infty}\nu(dd^cV,\varphi,t)$ is then
obvious.\qed
\endproof

\claim{(6.9) Example} \rm Let $X$ be an open subset of $\bC^n$ equipped with the
semi-exhaustive function $\varphi(z)=\log|z-a|$, $a\in X$. Then 
$(dd^c\varphi)^n=\delta_a$ and the Jensen-Lelong formula becomes
$$\mu_r(V)=V(a)+\int_{-\infty}^r\nu(dd^cV,\varphi,t)\,dt.$$
As $\mu_r$ is the mean value measure on the sphere $S(a,e^r)$, we
make the change of variables $r\mapsto\log r$, $t\mapsto\log t$ and
obtain the more familiar formula
$$\mu(V,S(a,r))=V(a)+\int_0^r\nu(dd^cV,a,t)\,{dt\over t}\leqno
(6.9\,{\rm a)}$$
where $\nu(dd^cV,a,t)=\nu(dd^cV,\varphi,\log t)$ is given by (5.7):
$$\nu(dd^cV,a,t)=
{1\over\pi^{n-1}t^{2n-2}/(n-1)!}\int_{B(a,t)}{1\over2\pi}\Delta V.
\leqno(6.9\,{\rm b)}$$
In this setting, Cor.~6.8 implies
$$\nu(dd^cV,a)=\lim_{r\to 0}{\mu\big(V,S(a,r)\big)\over\log r}=
\lim_{r\to 0}{\sup_{S(a,r)}V\over\log r}.\leqno(6.9\,{\rm c)}$$
To prove the last equality, we may assume $V\le 0$ after subtraction of
a constant. Inequality $\ge$ follows from the
obvious estimate $\mu(V,S(a,r))\le\sup_{S(a,r)}V$, while inequality $\le$
follows from the standard Harnack estimate
$$\sup_{S(a,\varepsilon r)}V\le{1-\varepsilon\over(1+\varepsilon)^{2n-1}}\,
\mu\big(V,S(a,r)\big)\leqno(6.9\,{\rm d)}$$
when $\varepsilon$ is small (this estimate follows easily from the Green-Riesz
representation formula 1.4.6 and 1.4.7). As $\sup_{S(a,r)}V=\sup_{B(a,r)}V$,
Formula (6.9$\,$c) can also be rewritten $\nu(dd^cV,a)=\liminf_{z\to a}
V(z)/\log|z-a|$. Since $\sup_{S(a,r)}V$ is a convex (increasing) function
of $\log r$, we infer that
$$V(z)\le\gamma\log|z-a|+{\rm O}(1)\leqno(6.9\,{\rm e)}$$
with $\gamma=\nu(dd^cV,a)$, and $\nu(dd^cV,a)$ is the largest constant
$\gamma$ which satisfies this inequa\-lity. Thus $\nu(dd^cV,a)=\gamma$
is equivalent to $V$ having a logarithmic pole of coefficient~$\gamma$.
\endclaim

\subsubsection{6.10}{Special case.} Take in particular $V=\log|f|$ where
$f$ is a holomorphic function on~$X$.
The Lelong-Poincar\'e formula shows that $dd^c\log|f|$ is
equal to the zero divisor $[Z_f]=\sum m_j[H_j]$, where $H_j$ are the
irreducible components of $f^{-1}(0)$ and $m_j$ is the multiplicity of
$f$ on $H_j$. The trace ${1\over 2\pi}\Delta\log|f|$ is then the euclidean
area measure of $Z_f$ (with corresponding multiplicities $m_j$).
By Formula (6.9$\,$c), we see that the Lelong number $\nu([Z_f],a)$
is equal to the vanishing order $\ord_a(f)$, that is, the smallest
integer $m$ such that $D^\alpha f(a)\ne 0$ for some multiindex $\alpha$
with~$|\alpha|=m$. In
dimension $n=1$, we have ${1\over2\pi}\Delta\log f=\sum m_j\delta_{a_j}$.
Then (6.9$\,a$) is the usual Jensen formula
$$\mu\big(\log|f|,S(0,r)\big)-\log|f(0)|=\int_0^r\nu(t){dt\over t}=
\sum m_j\log{r\over|a_j|}$$
where $\nu(t)$ is the number of zeros $a_j$ in the disk $D(0,t)$,
counted with multi\-plicities $m_j$.

\claim{(6.11) Example} \rm Take $\varphi(z)=\log\max|z_j|^{\lambda_j}$
where $\lambda_j>0$. Then $B(r)$ is the polydisk of
radii $(e^{r/\lambda_1},\ldots,e^{r/\lambda_n})$. If some coordinate $z_j$
is non zero, say $z_1$, we can write $\varphi(z)$ as
$\lambda_1\log|z_1|$ plus some function depending only on
the $(n-1)$ variables $z_j/z_1^{\lambda_1/\lambda_j}$. Hence
$(dd^c\varphi)^n=0$ on $\bC^n\setminus\{0\}$. It will be shown
later that 
$$(dd^c\varphi)^n=\lambda_1\ldots\lambda_n\,\delta_0.\leqno(6.11\,{\rm a})$$
We now determine the measures $\mu_r$.
At any point $z$ where not all terms $|z_j|^{\lambda_j}$
are equal, the smallest one can be omitted without changing $\varphi$
in a neighborhood of $z$. Thus $\varphi$ depends only on
$(n-1)$-variables and $(dd^c\varphi_{\ge r})^n=0$, $\mu_r=0$
near $z$. It follows that $\mu_r$ is supported by the
distinguished boundary $|z_j|=e^{r/\lambda_j}$ of the polydisk $B(r)$.
As $\varphi$ is invariant by all rotations $z_j\longmapsto
e^{\ii\theta_j}z_j$, the measure $\mu_r$ is also invariant and we see that
$\mu_r$ is a constant multiple of $d\theta_1\ldots d\theta_n$.
By formula (6.2) and (6.11$\,$a) we get
$$\mu_r=\lambda_1\ldots\lambda_n\,(2\pi)^{-n}d\theta_1\ldots d\theta_n.
\leqno(6.11\,{\rm b})$$
In particular, the Lelong number $\nu(dd^cV,\varphi)$ is given by
$$\nu(dd^cV,\varphi)=\lim_{r\to-\infty}{\lambda_1\ldots\lambda_n\over r}
\int_{\theta_j\in[0,2\pi]}\kern-6pt V(e^{r/\lambda_1+\ii\theta_1}
,\ldots,e^{r/\lambda_n+\ii\theta_n})\,{d\theta_1\ldots d\theta_n\over(2\pi)^n}.$$
These numbers have been introduced and studied by \cite{Kiselman 1986}.
We call them {\it directional Lelong numbers} with coefficients
$(\lambda_1,\ldots,\lambda_n)$. For an arbitrary current $T$, we define
$$\nu(T,x,\lambda)=\nu\big(T,\log\max|z_j-x_j|^{\lambda_j}\big).
\leqno(6.11\,{\rm c})$$
The above formula for $\nu(dd^cV,\varphi)$ combined with the analogue of
Harnack's inequality (6.9$\,$d) for polydisks gives
$$\leqalignno{
\nu(dd^cV,x,\lambda)&=\lim_{r\to 0}{\lambda_1\ldots\lambda_n\over\log r}
\int V(r^{1/\lambda_1}e^{\ii\theta_1},\ldots,r^{1/\lambda_n}
e^{\ii\theta_n})\,{d\theta_1\ldots d\theta_n\over(2\pi)^n}\cr
&=\lim_{r\to 0}{\lambda_1\ldots\lambda_n\over\log r}
\sup_{\theta_1,\ldots,\theta_n}V(r^{1/\lambda_1}e^{\ii\theta_1},\ldots,r^{1/\lambda_n}
e^{\ii\theta_n}).&(6.11\,{\rm d})\cr}$$
\endclaim

\section{7}{Comparison Theorems for Lelong Numbers}
Let $T$ be a closed positive current of bidimension $(p,p)$ on a Stein
manifold $X$ equipped with a semi-exhaustive plurisubharmonic
weight~$\varphi$. We first show that the Lelong numbers $\nu(T,\varphi)$ only
depend on the asymptotic behaviour of $\varphi$ near the polar set
$S(-\infty)$. In a precise way:

\claim{(7.1) First comparison theorem} Let
$\varphi,\psi:X\longrightarrow[-\infty,+\infty[$
be continuous plurisubharmonic functions. We assume that $\varphi,\psi$ are 
semi-exhaustive on $\Supp\,T$ and that
$$\ell:=\limsup{{\psi(x)}\over{\varphi(x)}}<+\infty~~~~{\it as}~~
x\in\Supp\,T~~{\it and}~~\varphi(x)\to -\infty.$$
Then $\nu(T,\psi)\le \ell^p\nu(T,\varphi)$, and the equality holds if
$\ell=\lim\psi/\varphi$.
\endclaim

\proof{Proof} Definition 6.4 shows immediately that
$\nu(T,\lambda\varphi)=\lambda^p\nu(T,\varphi)$
for every scalar $\lambda>0$. It is thus sufficient to verify the
inequality $\nu(T,\psi)\le\nu(T,\varphi)$ under the hypothesis 
$\limsup\psi/\varphi<1$. For all $c>0$, consider the plurisubharmonic
function 
$$u_c=\max(\psi-c,\varphi).$$
Let $R_\varphi$ and $R_\psi$
be such that $B_\varphi(R_\varphi)\cap\Supp\,T$ and
$B_\psi(R_\psi)\cap\Supp\,T$ be relatively compact in $X$.
Let $r<R_\varphi$ and $a<r$ be fixed. For $c>0$ large enough, we have 
$u_c=\varphi$ on $\varphi^{-1}([a,r])$ and Stokes' formula gives
$$\nu(T,\varphi,r)=\nu(T,u_c,r)\ge\nu(T,u_c).$$
The hypothesis $\limsup\psi/\varphi<1$ implies on the other hand that
there exists $t_0<0$ such that $u_c=\psi-c$ on $\{u_c<t_0\}\cap
\Supp\,T$. We infer 
$$\nu(T,u_c)=\nu(T,\psi-c)=\nu(T,\psi),$$
hence $\nu(T,\psi)\le\nu(T,\varphi)$. The equality case is obtained by
reversing the roles of $\varphi$ and $\psi$ and observing that
$\lim\varphi/\psi=1/l$.\qed
\endproof

  Assume in particular that $z^k=(z^k_1,\ldots,z^k_n)$, $k=1,2$, are
coordinate systems centered at a point $x\in X$ and let
   $$\varphi_k(z)=\log|z^k|=\log\bigl(|z^k_1|^2+\ldots+|z^k_n|^2\bigr)^{1/2}.$$
We have $\lim_{z\to x}{\varphi_2(z)/\varphi_1(z)}=1$, hence 
$\nu(T,\varphi_1)=\nu(T,\varphi_2)$ by Th.~7.1. 

\claim{(7.2) Corollary} The usual Lelong numbers $\nu(T,x)$ are independent
of the choice of local coordinates.\qed
\endclaim

This result had been originally proved by \cite{Siu 1974} with a much more
delicate proof. Another interesting consequence is:

\claim{(7.3) Corollary} On an open subset of $\bC^n$, the Lelong numbers
and Kiselman numbers are related by
     $$\nu(T,x)=\nu\big(T,x,(1,\ldots,1)\big).$$
\endclaim

\proof{Proof} By definition, the Lelong number $\nu(T,x)$ is associated with
the weight $\varphi(z)=\log|z-x|$ and the Kiselman number 
$\nu\big(T,x,(1,\ldots,1)\big)$ to the weight
$\psi(z)=\log\max|z_j-x_j|$. It is clear that
$\lim_{z\to x}\psi(z)/\varphi(z)=1$, whence the conclusion.\qed
\endproof

Another consequence of Th.~7.1 is that $\nu(T,x,\lambda)$ is an
increasing function of each variable~$\lambda_j$. Moreover, if
$\lambda_1\le\ldots\le\lambda_n$, we get the inequalities
$$\lambda_1^p\nu(T,x)\le\nu(T,x,\lambda)\le\lambda_n^p\nu(T,x).$$
These inequalities will be improved in section~7
(see Cor.~9.16). For the moment, we just prove the following
special case.

\claim{(7.4) Corollary} For all $\lambda_1,\ldots,\lambda_n>0$ we have
$$\big(dd^c\log\max_{1\le j\le n}|z_j|^{\lambda_j}\big)^n=
\Big(dd^c\log\sum_{1\le j\le n}|z_j|^{\lambda_j}\Big)^n=
\lambda_1\ldots\lambda_n\,\delta_0.$$
\endclaim

\proof{Proof} In fact, our measures vanish on $\bC^n\ssm\{0\}$ by the arguments
explained in example~6.11. Hence they are equal to $c\,\delta_0$
for some constant $c\ge 0$ which is simply the Lelong number of the
bidimension $(n,n)$-current $T=[X]=1$ with the corresponding weight.
The comparison theorem shows that the first equality holds and that
$$\Big(dd^c\log\sum_{1\le j\le n}|z_j|^{\lambda_j}\Big)^n=
\ell^{-n}\Big(dd^c\log\sum_{1\le j\le n}|z_j|^{\ell\lambda_j}\Big)^n$$
for all $\ell>0$. By taking $\ell$ large and approximating $\ell\lambda_j$
with $2[\ell\lambda_j/2]$, we may assume that $\lambda_j=2s_j$ is
an even integer. Then formula (5.6) gives
$$\eqalign{
&\int_{\sum|z_j|^{2s_j}<r^2}\Big(dd^c\log\sum|z_j|^{2s_j}\Big)^n=
r^{-2n}\int_{\sum|z_j|^{2s_j}<r^2}\Big(dd^c\sum|z_j|^{2s_j}\Big)^n\cr
&\qquad{}=s_1\ldots s_n\,r^{-2n}\int_{\sum|w_j|^2<r^2}2^n\Big({\ii\over 2\pi}
d'd''|w|^2\Big)^n=\lambda_1\ldots\lambda_n\cr}$$
by using the $s_1\ldots s_n$-sheeted change of variables
$w_j=z_j^{s_j}$.\qed
\endproof


Now, we assume that $T=[A]$ is the current of integration over an
analytic set $A\subset X$ of pure dimension~$p$.
The above comparison theorem will enable us to give a simple proof
of P.~Thie's main result \cite{Thie~1967}: the Lelong number $\nu([A],x)$ 
can be interpreted as the multiplicity of the analytic set $A$ at
point~$x$. Our starting point is the following consequence of
Th.~II.3.19 applied simultaneously to all irreducible components of~$(A,x)$.

\claim{(7.5) Lemma} For a generic choice of local coordinates 
$z'=(z_1,\ldots,z_p)$ and\break $z''=(z_{p+1},\ldots,z_n)$
on $(X,x)$, the germ $(A,x)$ is contained
in a cone \hbox{$|z''|\le C|z'|$}. If
$B'\subset\bC^p$ is a ball of center $0$ and radius $r'$ small,
and \hbox{$B''\subset\bC^{n-p}$} is the ball of center $0$ and
radius~$r''=Cr'$, then the projection
      $${\rm pr}:A\cap(B'\times B'')\longrightarrow B'$$
is a ramified covering with finite sheet number $m$.\qed
\endclaim

We use these properties to compute the Lelong number
of $[A]$ at point~$x$. When $z\in A$ tends to $x$, the functions
$$\varphi(z)=\log|z|=\log(|z'|^2+|z''|^2)^{1/2},~~~~\psi(z)=\log|z'|.$$
are equivalent. As $\varphi,\psi$ are semi-exhaustive on $A$, 
Th.~7.1 implies
        $$\nu([A],x)=\nu([A],\varphi)=\nu([A],\psi).$$
Let us apply formula (5.6) to $\psi\,$: for every $t<r'$ we get
$$\eqalign{\nu([A],\psi,\log t)
&=t^{-2p}\int_{\{\psi<\log t\}}[A]\wedge\Big({1\over 2}dd^ce^{2\psi}\Big)^p\cr
&=t^{-2p}\int_{A\cap\{|z'|<t\}}\Big({1\over 2}{\rm pr}^\star dd^c|z'|^2\Big)^p\cr
&=m\,t^{-2p}\int_{\bC^p\cap\{|z'|<t\}}\Big({1\over 2}dd^c|z'|^2\Big)^p=m,\cr}$$
hence $\nu([A],\psi)=m$.  Here, we have used the fact that pr is an \'etale
covering with $m$ sheets over the complement of the ramification locus
$S\subset B'$, and the fact that $S$ is of zero Lebesgue measure in~$B'$.
We have thus obtained simultaneously the following two results:

\claim{(7.6) Theorem and Definition} Let $A$ be an analytic set
of dimension $p$ in a complex manifold $X$ of dimension~$n$. For a
generic choice of local coordinates $z'=(z_1,\ldots,z_p)$, $z''=(z_{p+1},\ldots,
z_n)$ near a point $x\in A$ such that the germ $(A,x)$ is
contained in a cone $|z''|\le C|z'|$, the sheet number $m$ of the
projection $(A,x)\to(\bC^p,0)$ onto the first $p$ coordinates is
independent of the choice of $z'$, $z''$. This number $m$ is called the
multiplicity of $A$ at~$x$.
\endclaim

\claim{(7.7) Theorem {\rm(\cite{Thie 1967})}} One has $\nu([A],x)=m$.\qed
\endclaim

There is another interesting version of the comparison theorem which
compares the Lelong numbers of two currents obtained as
intersection pro\-ducts (in that case, we take the same weight for both).

\claim{(7.8) Second comparison theorem} Let $u_1,\ldots,u_q$ and $v_1,\ldots,v_q$
be plurisubharmonic functions such that each $q$-tuple satisfies the
hypotheses of Th.~4.5 with respect to~$T$. Suppose moreover that
$u_j=-\infty$ on $\Supp\,T\cap\varphi^{-1}(-\infty)$ and that
$$\ell_j:=\limsup{v_j(z)\over u_j(z)}<+\infty~~~~\hbox{\rm when}~~
z\in\Supp\,T\ssm u_j^{-1}(-\infty),~~\varphi(z)\to -\infty.$$
Then
$$\nu(dd^cv_1\wedge\ldots\wedge dd^cv_q\wedge T,\varphi)\le
\ell_1\ldots\ell_q\,\nu(dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge T,\varphi).$$
\endclaim

\proof{Proof} By homogeneity in each factor $v_j$, it is enough to prove the
inequality with constants $\ell_j=1$ under the hypothesis $\limsup v_j/u_j<1$.
We set
$$w_{j,c}=\max\{v_j-c,u_j\}.$$
Our assumption implies that $w_{j,c}$ coincides with $v_j-c$ on a
neighborhood $\Supp\,T\cap\{\varphi<r_0\}$ of
$\Supp\,T\cap\{\varphi<-\infty\}$, thus
$$\nu(dd^cv_1\wedge\ldots\wedge dd^cv_q\wedge T,\varphi)=
\nu(dd^cw_{1,c}\wedge\ldots\wedge dd^cw_{q,c}\wedge T,\varphi)$$
for every $c$. Now, fix $r<R_\varphi$. Proposition~4.9 shows that the
current $dd^cw_{1,c}\wedge\ldots\wedge dd^cw_{q,c}\wedge T$ converges
weakly to $dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge T$ when $c$ tends
to~$+\infty$. By Prop.~5.12 we get
$$\limsup_{c\to+\infty}~\nu(dd^cw_{1,c}\wedge\ldots\wedge dd^cw_{q,c}\wedge T,
\varphi)\le\nu(dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge T,\varphi).\eqno\square$$
\endproof

\claim{(7.9) Corollary} If $dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge T$ is
well defined, then at every point $x\in X$ we have
$$\nu\big(dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge T,x\big)
  \ge\nu(dd^cu_1,x)\ldots\nu(dd^cu_q,x)\,\nu(T,x).$$
\endclaim

\proof{Proof} Apply (7.8) with $\varphi(z)=v_1(z)=\ldots=v_q(z)=\log|z-x|$ and
observe that $\ell_j:=\limsup v_j/u_j=1/\nu(dd^cu_j,x)$ (there is nothing
to prove if $\nu(dd^cu_j,x)=0$).\qed
\endproof

Finally, we present an interesting stability property of Lelong
numbers due to\break \cite{Siu~1974}: almost all slices of a closed
positive current $T$ along linear subspaces passing through a given
point have the same Lelong number as~$T$. Before giving a proof of
this, we need a useful formula known as {\it Crofton's formula}.

\claim{(7.10) Lemma} Let $\alpha$ be a closed positive $(p,p)$-form on
$\bC^n\ssm\{0\}$ which is invariant under the unitary group~$U(n)$.
Then $\alpha$ has the form
$$\alpha=\big(dd^c\chi(\log|z|)\big)^p$$
where $\chi$ is a convex increasing function. Moreover $\alpha$ is
invariant by homotheties if and only if $\chi$ is an affine function, i.e.\
$\alpha=\lambda\,(dd^c\log|z|)^p$.
\endclaim

\proof{Proof} A radial convolution $\alpha_\varepsilon(z)=\int_\bR
\rho(t/\varepsilon)\,\alpha(e^tz)\,dt$ produces a smooth form with the
same properties as~$\alpha$ and $\lim_{\varepsilon\to 0}\alpha_\varepsilon=
\alpha$. Hence we can suppose that $\alpha$ is smooth on $\bC^n\ssm\{0\}$.
At a point $z=(0,\ldots,0,z_n)$, the $(p,p)$-form $\alpha(z)\in\bigwedge^{p,p}
(\bC^n)^\star$ must be invariant by $U(n-1)$ acting on the first
$(n-1)$ coordinates. We claim that the subspace of $U(n-1)$-invariants
in $\bigwedge^{p,p}(\bC^n)^\star$ is generated by
$(dd^c|z|^2)^p$ and $(dd^c|z|^2)^{p-1}\wedge\ii dz_n\wedge d\ol z_n$.
In fact, a form $\beta=\sum\beta_{I,J}dz_I\wedge d\ol z_J$ is invariant
by $U(1)^{n-1}\subset U(n-1)$ if and only if $\beta_{I,J}=0$ for
$I\ne J$, and invariant by the permutation group
$\cS_{n-1}\subset U(n-1)$ if and only if all coefficients
$\beta_{I,I}$ (resp.\ $\beta_{Jn,Jn})$ with
$I,J\subset\{1,\ldots,n-1\}$ are equal. Hence
$$\beta=\lambda\sum_{|I|=p}dz_I\wedge d\ol z_I+\mu
\Big(\sum_{|J|=p-1}dz_J\wedge d\ol z_J\Big)\wedge dz_n\wedge d\ol z_n.$$
This proves our claim. As $d|z|^2\wedge d^c|z|^2={\ii\over\pi}|z_n|^2
dz_n\wedge d\ol z_n$ at $(0,\ldots,0,z_n)$, we conclude that
$$\alpha(z)=f(z)(dd^c|z|^2)^p+g(z)(dd^c|z|^2)^{p-1}\wedge
d|z|^2\wedge d^c|z|^2$$
for some smooth functions $f,g$ on $\bC^n\ssm\{0\}$.
The $U(n)$-invariance of $\alpha$ shows that $f$ and $g$ are radial functions.
We may rewrite the last formula as
$$\alpha(z)=u(\log|z|)(dd^c\log|z|)^p+v(\log|z|)(dd^c\log|z|)^{p-1}\wedge
d\log|z|\wedge d^c\log|z|.$$
Here $(dd^c\log|z|)^p$ is a positive $(p,p)$-form coming from $\bP^{n-1}$,
hence it has zero contraction in the radial direction, while the
contraction of the form $(dd^c\log|z|)^{p-1}\wedge d\log|z|\wedge d^c\log|z|$
by the radial vector field is $(dd^c\log|z|)^{p-1}$. This shows easily that
$\alpha(z)\ge 0$ if and only if $u,v\ge 0$. Next, the closedness
condition $d\alpha=0$ gives $u'-v=0$. Thus $u$ is increasing and we define
a convex increasing function $\chi$ by $\chi'=u^{1/p}$. Then
$v=u'=p\chi^{\prime p-1}\chi''$ and
$$\alpha(z)=\big(dd^c\chi(\log|z|)\big)^p.$$
If $\alpha$ is invariant by homotheties, the functions
$u$ and $v$ must be constant, thus $v=0$ and $\alpha=\lambda
(dd^c\log|z|)^p$.\qed
\endproof

\claim{(7.11) Corollary {\rm(Crofton's formula)}} Let $dv$ be the
unique $U(n)$-invariant measure of mass $1$ on the Grassmannian
$G(p,n)$ of $p$-dimensional subspaces in $\bC^n$. Then
$$\int_{S\in G(p,n)}[S]\,dv(S)=(dd^c\log|z|)^{n-p}.$$
\endclaim

\proof{Proof} The left hand integral is a closed positive bidegree $(n-p,n-p)$
current which is invariant by $U(n)$ and by homotheties.
By Lemma~7.10, this current must coincide with the form
$\lambda(dd^c\log|z|)^{n-p}$ for some $\lambda\ge 0$. The coefficient
$\lambda$ is the Lelong number at~$0$. As $\nu([S],0)=1$ for every~$S$,
we get $\lambda=\int_{G(p,n)}dv(S)=1$.\qed
\endproof

We now recall a few basic facts of slicing theory; see \cite{Federer 1969}
for details. Let $\sigma:M\to M'$ be a submersion of smooth differentiable
manifolds and let $\Theta$ be a {\it locally flat} current on~$M$,
that is a current which can be written locally as $\Theta=U+dV$
where $U$, $V$ have locally integrable coefficients. It can be shown
that every current $\Theta$ such that both $\Theta$ and $d\Theta$ have
measure coefficients is locally flat; in particular,
closed positive currents are locally flats. Then, for almost every~$x'\in M'$,
there is a well defined slice $\Theta_{x'}$,
which is the current on the fiber $\sigma^{-1}(x')$ defined by
$$\Theta_{x'}=U_{\restriction\sigma^{-1}(x')}+
  dV_{\restriction\sigma^{-1}(x')}.$$
The restrictions of $U$, $V$ to the fibers exist for almost all $x'$ by
the Fubini theorem. It is easy to show by a regularization
$\Theta_\varepsilon=\Theta\star\rho_\varepsilon$ that
the slices of a closed positive current are again closed and positive:
in fact $U_{\varepsilon,x'}$ and $V_{\varepsilon,x'}$ converge to $U_{x'}$
and $V_{x'}$ in $L^1_{\rm loc}$, thus
$\Theta_{\varepsilon,x'}$ converges weakly to $\Theta_{x'}$
for almost every~$x'$. This kind of slicing can be referred to as
{\it parallel slicing} (if~we think of $\sigma$ as being a projection map).
The kind of slicing we need (where~the slices are taken over linear subspaces
passing through a given point) is of a slightly different nature and is
called {\it concurrent slicing}.

The possibility of concurrent slicing is proved as follows. Let $T$ be a closed
positive current of bidimension $(p,p)$ in the ball $B(0,R)\subset\bC^n$.
Let
$$Y=\big\{(x,S)\in\bC^n\times G(q,n)\,;\,x\in S\big\}$$
be the total space of the tautological rank $q$ vector bundle over the
Grassmannian $G(q,n)$, equipped with the obvious projections
$$\sigma:Y\longrightarrow G(q,n),~~~~\pi:Y\longrightarrow\bC^n.$$
We set $Y_R=\pi^{-1}(B(0,R))$ and $Y_R^\star=\pi^{-1}
(B(0,R)\ssm\{0\})$. The restriction $\pi_0$ of $\pi$ to $Y_R^\star$ is a
submersion onto $B(0,R)\ssm\{0\}$, so we have a well defined pull-back
$\pi_0^\star T$ over $Y_R^\star$. We would like to extend it as
a pull-back $\pi^\star T$ over~$Y_R$, so as to define slices
$T_{\restriction S}=(\pi^\star T)_{\restriction\sigma^{-1}(S)}\,$;
of course, these slices can be non zero only if the dimension of $S$
is at least equal to the degree of~$T$, i.e.\ if $q\ge n-p$.
We first claim that $\pi_0^\star T$ has
locally finite mass near the zero section $\pi^{-1}(0)$ of~$\sigma$.
In fact let $\omega_G$ be a unitary invariant K\"ahler metric over
$G(q,n)$ and let $\beta=dd^c|z|^2$ in~$\bC^n$. Then we get a
K\"ahler metric on $Y$ defined by $\omega_Y=\sigma^\star\omega_G+
\pi^\star\beta$. If $N=(q-1)(n-q)$ is the dimension of the fibers
of~$\pi$, the projection formula $\pi_\star(u\wedge\pi^\star v)=
(\pi_\star u)\wedge v$ gives
$$\pi_\star\omega_Y^{N+p}=\sum_{1\le k\le p}{N+p\choose k}
\beta^k\wedge\pi_\star(\sigma^\star\omega_G^{N+p-k}).$$
Here $\pi_\star(\sigma^\star\omega_G^{N+p-k})$ is a unitary and homothety
invariant $(p-k,p-k)$ closed positive form on $\bC^n\ssm\{0\}$,
so $\pi_\star(\sigma^\star\omega_G^{N+p-k})$ is proportional to
$(dd^c\log|z|)^{p-k}$. With some constants $\lambda_k>0$, we thus get
$$\eqalign{
\int_{Y_r^\star}\pi_0^\star T\wedge\omega_Y^{N+p}
&=\sum_{0\le k\le p}\lambda_k\int_{B(0,r)\ssm\{0\}}T\wedge\beta^k\wedge
(dd^c\log|z|)^{p-k}\cr
&=\sum_{0\le k\le p}\lambda_k2^{-(p-k)}r^{-2(p-k)}
\int_{B(0,r)\ssm\{0\}}T\wedge\beta^p<+\infty.\cr}$$
The Skoda-El Mir theorem 2.3 shows that the trivial extension
$\wt\pi_0^\star T$ of $\pi_0^\star T$ is a closed positive current on~$Y_R$.
Of course, the zero section $\pi^{-1}(0)$ might also carry some extra
mass of the desired current~$\pi^\star T$. Since $\pi^{-1}(0)$ has
codimension~$q$, this extra mass cannot exist when
$q>n-p=\codim\pi^\star T$ and we simply set $\pi^\star T=\wt\pi_0^\star T$.
On the other hand, if $q=n-p$, we set
$$\pi^\star T:=\wt\pi_0^\star T+\nu(T,0)\,[\pi^{-1}(0)].\leqno(7.12)$$
We can now apply parallel slicing with respect to $\sigma:Y_R\to G(q,n)$,
which is a submersion: for almost all $S\in G(q,n)$, there is a
well defined slice $T_{\restriction S}=
(\pi^\star T)_{\restriction\sigma^{-1}(S)}$. These slices
coincide with the usual restrictions of $T$ to $S$ if $T$ is smooth. 

\claim{(7.13) Theorem {\rm (\cite{Siu~1974})}} For almost all $S\in G(q,n)$
with $q\ge n-p$, the slice $T_{\restriction S}$ satisfies~
$\nu(T_{\restriction S},0)=\nu(T,0)$.
\endclaim

\proof{Proof} If $q=n-p$, the slice $T_{\restriction S}$ consists of some positive
measure with support in $S\ssm\{0\}$ plus a Dirac
measure $\nu(T,0)\,\delta_0$ coming from the slice of
$\nu(T,0)\,[\pi^{-1}(0)]$. The equality $\nu(T_{\restriction S},0)=
\nu(T,0)$ thus follows directly from~(7.12). 

In the general case $q>n-p$, it is clearly sufficient to prove the
following two properties:
\smallskip
\noindent a)~~ $\nu(T,0,r)=\displaystyle\int_{S\in G(q,n)}
\nu(T_{\restriction S},0,r)\,dv(S)$~~ for all $r\in{}]0,R[\,$;
\smallskip
\noindent b)~~ $\nu(T_{\restriction S},0)\ge\nu(T,0)$~~ for almost all $S$.
\smallskip
\noindent In fact, a) implies that $\nu(T,0)$ is the average
of all Lelong numbers $\nu(T_{\restriction S},0)$ and the conjunction
with b) implies that these numbers must be equal to $\nu(T,0)$ for
almost all $S$. In order to prove a) and b), we can suppose without loss
of generality that $T$ is smooth on $B(0,R)\ssm\{0\}$. Otherwise,
we perform a small convolution with respect to the action of
${\rm Gl}_n(\bC)$ on $\bC^n$:
$$T_\varepsilon=\int_{g\in{\rm Gl}_n(\bC)}\rho_\varepsilon(g)\,
g^\star T\,dv(g)$$
where $(\rho_\varepsilon)$ is a regularizing family with support in
an $\varepsilon$-neighborhood of the unit element of ${\rm Gl}_n(\bC)$.
Then $T_\varepsilon$ is smooth in $B(0,(1-\varepsilon)R)\ssm\{0\}$ and
converges weakly to $T$. Moreover, we have
$\nu(T_\varepsilon,0)=\nu(T,0)$ by (7.2) and
$\nu(T_{\restriction S},0)\ge\limsup_{\varepsilon\to 0}\nu(T_{\varepsilon,
\restriction S},0)$ by (5.12), thus a), b) are preserved in the limit.
If $T$ is smooth on $B(0,R)\ssm\{0\}$, the slice $T_{\restriction S}$
is defined for all $S$ and is simply the restriction of $T$ to
$S\ssm\{0\}$ (carrying no mass at the origin).
\smallskip
\noindent a) Here we may even assume that $T$ is smooth at $0$ by
performing an ordinary convolution. As $T_{\restriction S}$ has
bidegree $(n-p,n-p)$, we have
$$\nu(T_{\restriction S},0,r)=\int_{S\cap B(0,r)}T\wedge\alpha_S^{q-(n-p)}
=\int_{B(0,r)}T\wedge [S]\wedge\alpha_S^{p+q-n}$$
where $\alpha_S=dd^c\log|w|$ and $w=(w_1,\ldots,w_q)$ are orthonormal
coordinates on~$S$. We simply have to check that
$$\int_{S\in G(q,n)}[S]\wedge\alpha_S^{p+q-n}\,dv(S)=(dd^c\log|z|)^p.$$
However, both sides are unitary and homothety invariant $(p,p)$-forms
with Lelong number $1$ at the origin, so they must coincide by Lemma~7.11.
\smallskip
\noindent b) We prove the inequality when $S=\bC^q\times\{0\}$.
By the comparison theorem~7.1, for every $r>0$ and
$\varepsilon>0$ we have
$$\leqalignno{
&\int_{B(0,r)}T\wedge\gamma_\varepsilon^p\ge\nu(T,0)~~~~\hbox{\rm where}
&(7.14)\cr
&\gamma_\varepsilon={1\over 2}dd^c\log(\varepsilon|z_1|^2+\ldots+
\varepsilon|z_q|^2+|z_{q+1}|^2+\ldots+|z_n|^2).\cr}$$
We claim that the current $\gamma_\varepsilon^p$ converges weakly to
$$[S]\wedge\alpha_S^{p+q-n}=
[S]\wedge\Big({1\over2}dd^c\log(|z_1|^2+\ldots+|z_q|^2)\Big)^{p+q-n}$$
as $\varepsilon$ tends to $0$. In fact, the Lelong number of
$\gamma_\varepsilon^p$ at $0$ is $1$, hence by homogeneity
$$\int_{B(0,r)}\gamma_\varepsilon^p\wedge(dd^c|z|^2)^{n-p}=(2r^2)^{n-p}$$
for all $\varepsilon,r>0$. Therefore the family $(\gamma_\varepsilon^p)$
is relatively compact in the weak topology. Since
$\gamma_0=\lim\gamma_\varepsilon$ is smooth on $\bC^n\ssm S$ and
depends only on $n-q$ variables ($n-q\le p$), we have
$\lim\gamma_\varepsilon^p=\gamma_0^p=0$ on $\bC^n\ssm S$. This
shows that every weak limit of $(\gamma_\varepsilon^p)$ has
support in $S$. Each of these is the direct image by inclusion
of a unitary and homothety invariant $(p+q-n,p+q-n)$-form
on $S$ with Lelong number equal to $1$ at $0$. Therefore we must have
$$\lim_{\varepsilon\to 0}\gamma_\varepsilon^p=(i_S)_\star
(\alpha_S^{p+q-n})=[S]\wedge\alpha_S^{p+q-n},$$
and our claim is proved (of course, this can also be checked by direct
elementary calculations). By taking the limsup in (7.14) we obtain
$$\nu(T_{\restriction S},0,r+0)=\int_{\ol B(0,r)}T\wedge[S]\wedge
\alpha_S^{p+q-n}\ge\nu(T,0)$$
(the singularity of $T$ at $0$ does not create any difficulty because
we can modify $T$ by a $dd^c$-exact form near $0$ to make it smooth
everywhere). Property b) follows when $r$ tends to $0$.\qed
\endproof

\section{8}{Siu's Semicontinuity Theorem}
  Let $X$, $Y$ be complex manifolds of dimension $n$, $m$ such that $X$ is
Stein. Let $\varphi:X\times Y\longrightarrow[-\infty,+\infty[$ be a continuous 
plurisubharmonic function. We assume that $\varphi$ is {\it semi-exhaustive}
with respect to $\Supp\,T$ , i.e.\ that for every compact subset 
$L\subset Y$ there exists $R=R(L)<0$ such that
 $$\{(x,y)\in\Supp\,T\times L\,;
            \,\varphi(x,y)\le R\}\compact X\times Y.\leqno(8.1)$$
Let $T$ be a closed positive current of bidimension $(p,p)$ on $X$. 
For every point $y\in Y$, the function $\varphi_y(x):=\varphi(x,y)$ 
is semi-exhaustive on $\Supp\,T\,$; one can therefore associate with $y$ 
a generalized Lelong number $\nu(T,\varphi_y)$. Proposition~5.13 implies
that the map $y\mapsto\nu(T,\varphi_y)$ is upper semi-continuous,
hence the upperlevel sets
$$E_c=E_c(T,\varphi)=\{y\in Y ;\nu(T,\varphi_y)\ge c\}\ ,\ c>0\leqno(8.2)$$
are closed. Under mild additional hypotheses, we are going to show
that the sets $E_c$ are in fact analytic subsets of~$Y$, following 
\cite{Demailly 1987a}. 

\claim{(8.3) Definition} We say that a function $f(x,y)$ is locally
H\"older continuous with respect to $y$ on $X\times Y$
if every point of $X\times Y$ has a neighborhood $\Omega$ on which
$$|f(x,y_1)-f(x,y_2)|\le M |y_1-y_2|^\gamma$$
for all $(x,y_1)\in\Omega$, $(x,y_2)\in\Omega$,
with some constants $M>0$, $\gamma\in{}]0,1]$, and suitable
coordinates on $Y$.
\endclaim

\claim{(8.4) Theorem {\rm(\cite{Demailly 1987a})}} Let $T$ be a closed
positive current on $X$ and let
       $$\varphi:X\times Y\longrightarrow[-\infty,+\infty[$$
be a continuous plurisubharmonic function. Assume that $\varphi$ is
semi-exhaustive on $\Supp\,T$ and that $e^{\varphi(x,y)}$ is locally
H\"older continuous with respect to $y$ on~$X\times Y$. Then the upperlevel
sets
          $$E_c(T,\varphi)=\{y\in Y;\nu(T,\varphi_y)\ge c\}$$
are analytic subsets of $Y$.
\endclaim

This theorem can be rephrased by saying that $y\longmapsto\nu(T,\varphi_y)$
is upper semi-conti\-nuous with respect to the analytic Zariski topology.
As a special case, we get the following important result of 
\cite{Siu~1974}:

\claim{(8.5) Corollary} If $T$ is a closed positive current of bidimension
$(p,p)$ on a complex manifold~$X$, the upperlevel sets
$E_c(T)=\{x\in X\,;\,\nu(T,x)\ge c\}$ of the usual Lelong numbers are
analytic subsets of dimension $\le p$.
\endclaim

\proof{Proof} The result is local, so we may assume that $X\subset\bC^n$ is an open
subset. Theorem~8.4 with $Y=X$ and
$\varphi(x,y)=\log|x-y|$ shows that $E_c(T)$ is analytic. Moreover,
Prop.~5.11 implies $\dim E_c(T)\le p$.\qed
\endproof

\noindent{\bf(8.6) Generalization.} Theorem 8.4 can be applied more
generally to weight functions of the type
  $$\varphi(x,y)=\max_j~\log\Big(\sum_k|F_{j,k}(x,y)|^{\lambda_{j,k}}\Big)$$
where $F_{j,k}$ are holomorphic functions on $X\times Y$ and where
$\gamma_{j,k}$ are positive real constants; in this case $e^\varphi$ is
H\"older continuous of exponent $\gamma=\min\{\lambda_{j,k},1\}$ and
$\varphi$ is semi-exhaustive with respect to the whole space $X$ as soon
as the projection
${\rm pr}_2:\bigcap F_{j,k}^{-1}(0)\longrightarrow Y$ is proper and finite.

For example, when $\varphi(x,y)=\log\max|x_j-y_j|^\lambda_j$ on
an open subset $X$ of $\bC^n$ , we see that the upperlevel sets for
Kiselman's numbers $\nu(T,x,\lambda)$ are analytic in $X$ (a result
first proved in \cite{Kiselman~1986}. More generally,
set $\psi_\lambda(z)=\log\max|z_j|^{\lambda_j}$
and $\varphi(x,y,g)=\psi_\lambda\big(g(x-y)\big)$ where $x,y\in\bC^n$
and $g\in{\rm Gl}(\bC^n)$. Then $\nu(T,\varphi_{y,g})$ 
is the Kiselman number of $T$ at $y$ when the coordinates have been rotated
by $g$. It is clear that $\varphi$ is plurisubharmonic in $(x,y,g)$ and
semi-exhaustive with respect to $x$, and that $e^\varphi$ is locally H\"older
continuous with respect to $(y,g)$. Thus the upperlevel sets
$$E_c=\{(y,g)\in X\times{\rm Gl}(\bC^n)\,;\,\nu(T,\varphi_{y,g})\ge c\}$$
are analytic in $X\times{\rm Gl}(\bC^n)$. 
However this result is not meaningful on a manifold,
because it is not invariant under coordinate changes. One can obtain
an invariant version as follows. Let $X$ be a manifold and let $J^k\cO_X$
be the bundle of $k$-jets of holomorphic functions on~$X$. We consider
the bundle $S_k$ over $X$ whose fiber $S_{k,y}$ is the set of $n$-tuples
of $k$-jets $u=(u_1,\ldots,u_n)\in(J^k\cO_{X,y})^n$ such that $u_j(y)=0$ and
$du_1\wedge\ldots\wedge du_n(y)\ne 0$. Let $(z_j)$ be local coordinates on
an open set $\Omega\subset X$. Modulo $O(|z-y|^{k+1})$, we can write 
$$u_j(z)=\sum_{1\le|\alpha|\le k}a_{j,\alpha}(z-y)^\alpha$$
with ${\rm det}(a_{j,(0,\ldots,1_k,\ldots,0)})\ne 0$. The numbers
$((y_j),(a_{j,\alpha}))$ define a coordinate system on the total space of
$S_{k\,\restriction\Omega}$. For $(x,(y,u))\in X\times S_k$, we introduce
the function
$$\varphi(x,y,u)=\log\max|u_j(x)|^{\lambda_j}=
\log\max\Big|\sum_{1\le|\alpha|\le k}a_{j,\alpha}(x-y)^\alpha
\Big|^{\lambda_j}$$
which has all properties required by Th.~8.4 on a neighborhood of the diagonal
$x=y$, i.e.\ a neighborhood of $X\times_X S_k$ in $X\times S_k$. For
$k$ large, we claim that Kiselman's directional Lelong numbers
$$\nu(T,y,u,\lambda):=\nu(T,\varphi_{y,u})$$
with respect to the coordinate system $(u_j)$ at $y$
do not depend on the selection of the jet representives and are therefore
canonically defined on~$S_k$. In fact, a change of $u_j$ by $O(|z-y|^{k+1})$
adds $O(|z-y|^{(k+1)\lambda_j})$ to $e^\varphi$, and we have $e^\varphi\ge
O(|z-y|^{\max\lambda_j})$. Hence by (7.1) it is enough to take
$k+1\ge\max\lambda_j/\min\lambda_j$. Theorem~8.4 then shows that the
upperlevel sets $E_c(T,\varphi)$ are analytic in~$S_k$.\qed
\medskip

\noindent{\it Proof of the Semicontinuity Theorem~8.4.}
As the result is local on $Y$, we may assume without loss of
generality that $Y$ is a ball in $\bC^m$. After addition of 
a constant to $\varphi$, we may also assume that
there exists a compact subset $K\subset X$ such that
$$\{(x,y)\in X\times Y ;\varphi(x,y)\le 0\}\subset K\times Y.$$
By Th.~7.1, the Lelong numbers depend only on the
asymptotic behaviour of $\varphi$ near the (compact) polar set 
$\varphi^{-1}(-\infty)\cap({\rm Supp T}\times Y)$. We can add 
a smooth strictly plurisubharmonic function on $X\times Y$ to make
$\varphi$ strictly plurisuharmonic. Then Richberg's approximation
theorem for continuous plurisubharmonic functions shows that there exists
a smooth plurisubharmonic function $\wt\varphi$ such that
$\varphi\le\wt\varphi\le\varphi+1$. We may therefore assume that
$\varphi$ is smooth on $(X\times Y)\setminus\varphi^{-1}(-\infty)$.
\medskip
\noindent $\bullet$ {\bf First step}: 
{\it construction of a local plurisubharmonic potential.}

   Our goal is to generalize the usual construction of plurisubharmonic
potentials associated with a closed positive current (\cite{Lelong 1967},
\cite{Skoda 1972a}). We replace here the usual kernel
$|z-\zeta|^{-2p}$ arising from the hermitian metric of $\bC^n$
by a kernel depending on the weight $\varphi$.
Let $\chi\in \cC^\infty(\bR,\bR)$ be an increasing function such that
$\chi(t)=t$ for $t\le-1$ and $\chi(t)=0$ for $t\ge 0$. We consider
the half-plane $H=\{z\in\bC\,;\,{\rm Re}z<-1\}$ and associate with
$T$ the potential function $V$ on $Y\times H$ defined by
$$V(y,z)=-\int^0_{{\rm Re}z}\nu(T,\varphi_y,t)\chi'(t)\,dt.\leqno(8.7)$$
For every $t>\Re z$, Stokes' formula gives
$$\nu(T,\varphi_y,t)=
       \int_{\varphi(x,y)<t}T(x)\wedge(dd^c_x\wt\varphi(x,y,z))^p$$
with $\wt\varphi(x,y,z):=\max\{\varphi(x,y),{\rm Re}z\}$. The Fubini
theorem applied to (8.7) gives
$$\eqalign{V(y,z)
&=-\int_{{\scriptstyle x\in X,\varphi(x,y)<t}\atop{\scriptstyle\Re z<t<0}}
 T(x)\wedge(dd^c_x\wt\varphi(x,y,z))^p~\chi'(t)dt\cr
&=\int_{x\in X}T(x)\wedge\chi(\wt\varphi(x,y,z))\,(dd^c_x\wt\varphi(x,y,z))^p.
\cr}$$
For all $(n-1,n-1)$-form $h$ of class $C^{\infty}$ with compact support 
in $Y\times H$, we get
$$\eqalign{
\langle dd^cV,h\rangle&=\langle V,dd^ch\rangle\cr
&=\int_{X\times Y\times H}T(x)\wedge\chi(\wt\varphi(x,y,z))
(dd^c\wt\varphi(x,y,z))^p\wedge dd^ch(y,z).\cr}$$
Observe that the replacement of $dd^c_x$ by the total differentiation
$dd^c=dd^c_{x,y,z}$ does not modify the integrand, because the
terms in $dx$, $d\ol x$ must have total bidegree $(n,n)$.
The current $T(x)\wedge\chi(\wt\varphi(x,y,z))h(y,z)$ has compact support in
$X\times Y\times H$. An integration by parts can thus be performed
to obtain
$$\langle dd^cV,h\rangle=\int_{X\times Y\times H}T(x)\wedge
  dd^c(\chi\circ\wt\varphi(x,y,z))\wedge(dd^c\wt\varphi(x,y,z))^p\wedge
  h(y,z).
$$
On the corona $\{-1\le\varphi(x,y)\le0\}$ we have $\wt\varphi(x,y,z)
=\varphi(x,y)$, whereas for $\varphi(x,y)<-1$ we get
$\wt\varphi<-1$ and $\chi\circ\wt\varphi=\wt\varphi$. As $\wt\varphi$
is plurisubharmonic, we see that $dd^cV(y,z)$ is the sum of the positive $(1,1)$-form
   $$(y,z)\longmapsto\int_{\{x\in X;\varphi(x,y)<-1\}}
                      T(x)\wedge(dd^c_{x,y,z}\wt\varphi(x,y,z))^{p+1}$$
and of the $(1,1)$-form independent of $z$
   $$y\longmapsto\int_{\{x\in X;-1\le\varphi(x,y)\le0\}}T\wedge 
                 dd^c_{x,y}(\chi\circ\varphi)\wedge(dd^c_{x,y}\varphi)^p.$$
As $\varphi$ is smooth outside $\varphi^{-1}(-\infty)$, this last
form has locally bounded coefficients. Hence $dd^cV(y,z)$ is $\ge 0$ except
perhaps for locally bounded terms. In addition, $V$ is
continuous on $Y\times H$ because $T\wedge(dd^c\wt\varphi_{y,z})^p$
is weakly continuous in the variables $(y,z)$ by Th.~3.5. 
We therefore obtain the following result.

\claim{(8.8) Proposition} There exists a positive plurisubharmonic
function $\rho$ in $\cC^\infty(Y)$ such that $\rho(y)+V(y,z)$ is plurisubharmonic
on $Y\times H$.
\endclaim

If we let $\Re z$ tend to $-\infty$, we see that the function
    $$U_0(y)=\rho(y)+V(y,-\infty)
            =\rho(y)-\int_{-\infty}^0\nu(T,\varphi_y,t)\chi'(t)dt$$
is locally plurisubharmonic or $\equiv-\infty$ on $Y$. Furthermore,
it is clear that \hbox{$U_0(y)=-\infty$} at every point $y$ such that
$\nu(T,\varphi_y)>0$. If $Y$ is connected and $U_0\not\equiv-\infty$,
we already conclude that the density set $\bigcup_{c>0}E_c$
is pluripolar in~$Y$.
\medskip
\noindent $\bullet$ {\bf Second step}: 
{\it application of Kiselman's minimum principle.}

Let $a\ge 0$ be arbitrary. The function
     $$Y\times H\ni(y,z)\longmapsto\rho(y)+V(y,z)-a{\rm Re}z$$
is plurisubharmonic and independent of ${\rm Im}\,z$. By Kiselman's
theorem~1.7.8, the Legendre transform
     $$U_a(y)=\inf_{r<-1}\big\{\rho(y)+V(y,r)-ar\big\}$$
is locally plurisubharmonic or $\equiv-\infty$ on $Y$.

\claim{(8.9) Lemma} Let $y_0\in Y$ be a given point.
\medskip
\item{\rm a)} If $a>\nu(T,\varphi_{y_0})$, then $U_a$ is bounded
below on a neighborhood of $y_0$.
\medskip
\item{\rm b)} If $a<\nu(T,\varphi_{y_0})$, then $U_a(y_0)=-\infty$.
\endclaim

\proof{Proof} By definition of $V$ (cf.\ $(8.7)$) we have
$$V(y,r)\le-\nu(T,\varphi_y,r)\int_r^0\chi'(t)dt=r\nu(T,\varphi_y,r)
                 \le r\nu(T,\varphi_y).\leqno(8.10)$$
Then clearly $U_a(y_0)=-\infty$ if $a<\nu(T,\varphi_{y_0})$. On the
other hand, if $\nu(T,\varphi_{y_0})<a$, there exists $t_0<0$ 
such that $\nu(T,\varphi_{y_0},t_0)<a$. Fix $r_0<t_0$. 
The semi-continuity property (5.13) shows that there exists a
neighborhood $\omega$ of $y_0$ such that $\sup_{y\in\omega}~
\nu(T,\varphi_y,r_0)<a$. For all $y\in\omega$, we get
$$V(y,r)\ge-C-a\int_r^{r_0}\chi'(t)dt=-C+a(r-r_0),$$
and this implies $U_a(y)\ge-C-a r_0$.\qed
\endproof

\claim{(8.11) Theorem} If $Y$ is connected and if $E_c\ne Y$, then $E_c$
is a closed complete pluripolar subset of $Y$, i.e.\ there exists
a continuous plurisubharmonic function $w:Y\longrightarrow[-\infty,+\infty[$ such that
$E_c=w^{-1}(-\infty)$.
\endclaim

\proof{Proof} We first observe that the family $(U_a)$ is increasing in $a$,
that $U_a=-\infty$ on $E_c$ for all $a<c$ and that
$\sup_{a<c}U_a(y)>-\infty$ if $y\in Y\setminus E_c$
(apply Lemma~8.9). For any integer $k\ge 1$, let $w_k\in \cC^\infty(Y)$ be a
plurisubharmonic regularization of $U_{c-1/k}$ such that
$w_k\ge U_{c-1/k}$ on $Y$ and $w_k\le -2^k$ on
$E_c\cap Y_k$ where $Y_k=\{y\in Y\,;\,d(y,\partial Y)\ge 1/k\}$. Then
Lemma~8.9~a) shows that the family
$(w_k)$ is uniformly bounded below on every compact subset of
$Y\setminus E_c$. We can also choose $w_k$ uniformly
bounded above on every compact subset of $Y$ because
$U_{c-1/k}\le U_c$. The function
$$w=\sum_{k=1}^{+\infty}~2^{-k}w_k$$
satifies our requirements.\qed
\endproof

\noindent $\bullet$ {\bf Third step}: 
{\it estimation of the singularities of the potentials} $U_a$.

\claim{(8.12) Lemma} Let $y_0\in Y$ be a given point, $L$ a compact
neighborhood of $y_0$, $K\subset X$ a compact subset and $r_0$
a real number $<-1$ such that
    $$\{(x,y)\in X\times L;\varphi(x,y)\le r_0\}\subset K\times L.$$
Assume that $e^{\varphi(x,y)}$ is locally H\"older continuous in $y$
and that
$$|f(x,y_1)-f(x,y_2)|\le M |y_1-y_2|^\gamma$$
for all $(x,y_1,y_2)\in K\times L\times L$. Then, for all 
$\varepsilon\in{}]0,1[$, there exists a real number
$\eta(\varepsilon)>0$ such that all $y\in Y$ with 
$|y-y_0|<\eta(\varepsilon)$ satisfy
    $$U_a(y)\le\rho(y)+\big((1-\varepsilon)^p\nu(T,\varphi_{y_0})-a\big)
        \Big(\gamma\log|y-y_0|+\log{{2eM}\over\varepsilon}\Big).$$
\endclaim

\proof{Proof} First, we try to estimate $\nu(T,\varphi_y,r)$ when $y\in L$ is
near $y_0$. Set
$$\left\{ 
\eqalign{\psi(x)&=(1-\varepsilon)\varphi_{y_0}(x)+\varepsilon r-\varepsilon/2\cr
         \psi(x)&=\max\bigl(\varphi_y(x),(1-\varepsilon)\varphi_{y_0}(x)+
                  \varepsilon r-\varepsilon/2\bigr)\cr
         \psi(x)&=\varphi_y(x)\cr}
\eqalign{&{~\rm if~}\cr
         &{~\rm if~}\cr
         &{~\rm if~}\cr}
\eqalign{&\varphi_{y_0}(x)\le r-1\cr
   r-1\le&\varphi_{y_0}(x)\le r\cr
     r\le&\varphi_{y_0}(x)\le r_0\cr}\right.$$
and verify that this definition is coherent when $|y-y_0|$ is small enough.
By hypothesis
$$|e^{\varphi_y(x)}-e^{\varphi_{y_0}(x)}|\le M|y-y_0|^\gamma.$$
This inequality implies
$$\eqalign{\varphi_y(x)&\le\varphi_{y_0}(x)+\log\bigl(1+M|y-y_0|^\gamma 
                          e^{-\varphi_{y_0}(x)}\bigr)\cr
           \varphi_y(x)&\ge\varphi_{y_0}(x)+\log\bigl(1-M|y-y_0|^\gamma
                          e^{-\varphi_{y_0}(x)}\bigr).\cr}$$
In particular, for $\varphi_{y_0}(x)=r$, we have
$(1-\varepsilon)\varphi_{y_0}(x)+\varepsilon r-\varepsilon/2=r-\varepsilon/2$,
thus
      $$\varphi_y(x)\ge r+\log(1-M|y-y_0|^\gamma e^{-r}).$$
Similarly, for $\varphi_{y_0}(x)=r-1$, we have
$(1-\varepsilon)\varphi_{y_0}(x)+\varepsilon r-\varepsilon/2=
r-1+\varepsilon/2$, thus
      $$\varphi_y(x)\le r-1+\log(1+M|y-y_0|^\gamma e^{1-r}).$$
The definition of $\psi$ is thus coherent as soon as
$M|y-y_0|^\gamma e^{1-r}\le\varepsilon/2$ , i.e.
      $$\gamma\log|y-y_0|+\log{{2eM}\over\varepsilon}\le r.$$
In this case $\psi$ coincides with $\varphi_y$ on a neighborhood of 
$\{\psi=r\}$ , and with
         $$(1-\varepsilon)\varphi_{y_0}(x)+\varepsilon r-\varepsilon/2$$
on a neighborhood of the polar set $\psi^{-1}(-\infty)$. By Stokes'
formula applied to $\nu(T,\psi,r)$, we infer
  $$\nu(T,\varphi_y,r)=\nu(T,\psi,r)\ge\nu(T,\psi)=
                 (1-\varepsilon)^p\nu(T,\varphi_{y_0}).$$
From (8.10) we get $V(y,r)\le r\nu(T,\varphi_y,r)$, hence
$$\leqalignno{
     U_a(y)&\le\rho(y)+V(y,r)-ar\le 
                   \rho(y)+r\big(\nu(T,\varphi_y,r)-a\big),\cr
     U_a(y)&\le\rho(y)+r\bigl((1-\varepsilon)^p\nu(T,\varphi_{y_0})-a\bigr).
                   &(8.13)\cr}$$
Suppose $\gamma\log|y-y_0|+\log(2eM/\varepsilon)\le r_0$ , i.e.\ 
$|y-y_0|\le(\varepsilon/2eM)^{1/\gamma}e^{r_0/\gamma}\,$; 
one can then choose $r=\gamma\log|y-y_0|+\log(2eM/\varepsilon)$, and by
$(8.13)$ this yields the inequality asserted in Th.~8.12.\qed
\endproof

\medskip
\noindent $\bullet$ {\bf Fourth step}:
{\it application of the H\"ormander-Bombieri-Skoda theorem.}

The end of the proof relies on the following crucial result, which is
a consequence of the H\"ormander-Bombieri-Skoda theorem (\cite{Bombieri~1970},
\cite{Skoda~1972a}, \cite{Skoda~1976}); it will be proved in 
Chapter~8, see Cor.~8.?.?.

\claim{(8.14) Proposition} Let $u$ be a plurisubharmonic function
on a complex  mani\-fold~$Y$. The set of points in a neighborhood of
which $e^{-u}$ is not integrable is an analytic subset of $Y$.\qed
\endclaim

\proof{Proof of Theorem~8.4 (end).} The main idea in what follows is due to
\cite{Kiselman 1979}. For $a,b>0$, we let $Z_{a,b}$ be the set of points
in a neighborhood of which $\exp(-U_a/b)$ is not integrable. Then
$Z_{a,b}$ is analytic, and as the family $(U_a)$ is increasing in $a$,
we have $Z_{a',b'}\supset Z_{a'',b''}$ if $a'\le a''$, $b'\le b''$.

Let $y_0\in Y$ be a given point. If $y_0\notin E_c$, then $\nu
(T,\varphi_{y_0})<c$ by definition of $E_c$. Choose $a$ such that $\nu
(T,\varphi_{y_0})<a<c$.
Lemma~8.9~a) implies that $U_a$ is bounded below in a neighborhood of
$y_0$, thus $\exp(-U_a/b)$ is integrable and $y_0\notin Z_{a,b}$ for all
$b>0$.

On the other hand, if $y_0\in E_c$ and if $a<c$, then Lemma~8.12 implies
for all $\varepsilon>0$ that
$$U_a(y)\le(1-\varepsilon)(c-a)\gamma\log|y-y_0|+C(\varepsilon)$$
on a neighborhood of $y_0$. Hence $\exp(-U_a/b)$ is non integrable at
$y_0$ as soon as $b<(c-a)\gamma/2m$, where $m=\dim Y$. We obtain
therefore
$$E_c=\bigcap_{{\scriptstyle a<c}\atop{\scriptstyle b<(c-a)\gamma/2m}}
Z_{a,b}.$$
This proves that $E_c$ is an analytic subset of $Y$.\qed
\endproof

Finally, we use Cor.~8.5 to derive an important decomposition
formula for currents, which is again due to \cite{Siu 1974}. We first begin
by two simple observations.

\claim{(8.15) Lemma} If $T$ is a closed
positive current of bidimension $(p,p)$ and $A$ is an irreducible
analytic set in $X$, we set
$$m_A=\inf\{\nu(T,x)\,;\,x\in A\}.$$
Then $\nu(T,x)=m_A$ for all $x\in A\ssm\bigcup A'_j$, where
$(A'_j)$ is a countable family of proper analytic subsets of $A$.
We say that $m_A$ is the generic Lelong number of $T$ along $A$.
\endclaim

\proof{Proof} By definition of $m_A$ and $E_c(T)$, we have $\nu(T,x)\ge m_A$ for
every $x\in A$ and
$$\nu(T,x)=m_A~~~~\hbox{\rm on}~~A\ssm\bigcup_{c\in\bQ,\,c>m_A}A\cap E_c(T).$$
However, for $c>m_A$, the intersection $A\cap E_c(T)$ is a proper analytic
subset of~$A$.\qed
\endproof

\claim{(8.16) Proposition} Let $T$ be a closed positive current of
bidimension $(p,p)$ and let $A$ be an irreducible $p$-dimensional
analytic subset of $X$. Then \hbox{$\bOne_AT=m_A[A]$,} in particular
$T-m_A[A]$ is positive.
\endclaim

\proof{Proof} As the question is local and as
a closed positive current of bidimension $(p,p)$ cannot
carry any mass on a $(p-1)$-dimensional analytic subset, it is
enough to work in a neighborhood of a regular point $x_0\in A$.
Hence, by choosing suitable coordinates, we can suppose that $X$
is an open set in $\bC^n$ and that $A$ is the intersection of
$X$ with a $p$-dimensional linear subspace. Then, for every point
$a\in A$, the inequality $\nu(T,a)\ge m_A$ implies
$$\sigma_T\big(B(a,r)\big)\ge m_A\,\pi^pr^{2p}/p!=
m_A\sigma_{[A]}\big(B(a,r)\big)$$
for all $r$ such that $B(a,r)\subset X$. Now, set $\Theta=T-m_A[A]$
and $\beta=dd^c|z|^2$.
Our inequality says that $\int\bOne_{B(a,r)}\Theta\wedge\beta^p\ge 0$.
If we integrate this with respect to some positive continuous
function $f$ with compact support in $A$, we get
$\int_X g_r\Theta\wedge\beta^p\ge 0$ where
$$g_r(z)=\int_A\bOne_{B(a,r)}(z)\,f(a)\,d\lambda(a)
=\int_{a\in A\cap B(z,r)}f(a)\,d\lambda(a).$$
Here $g_r$ is continuous on $\bC^n$, and as $r$ tends to $0$ the function
$g_r(z)/(\pi^pr^{2p}/p!)$ converges to $f$ on $A$ and to $0$
on $X\ssm A$, with a global
uniform bound. Hence we obtain $\int\bOne_Af\,\Theta\wedge\beta^p\ge 0$.
Since this inequality is true for all continuous functions $f\ge 0$ with
compact support in $A$, we conclude that the measure
$\bOne_A\Theta\wedge\beta^p$ is positive. By a linear change of coordinates,
we see that
$$\bOne_A\Theta\wedge\Big(dd^c\sum_{1\le j\le n}
\lambda_j|u_j|^2\Big)^n\ge 0$$
for every basis $(u_1,\ldots,u_n)$ of linear forms and for all
coefficients~\hbox{$\lambda_j>0$.}
Take $\lambda_1=\ldots=\lambda_p=1$ and let the other
$\lambda_j$ tend to~$0$. Then we get
$$\bOne_A\Theta\wedge\ii du_1\wedge d\ol u_1\wedge\ldots\wedge
du_p\wedge d\ol u_p\ge 0.
$$
This implies $\bOne_A\Theta\ge 0$,
or equivalently $\bOne_AT\ge m_A[A]$. By Cor.~2.4
we know that $\bOne_AT$ is a closed positive current, thus
$\bOne_AT=\lambda[A]$ with $\lambda\ge 0$. We have just seen
that $\lambda\ge m_A$. On the other hand, $T\ge\bOne_AT=\lambda[A]$
clearly implies $m_A\ge\lambda$.\qed
\endproof

\claim{(8.16) Siu's decomposition formula} If $T$ is a closed
positive current of bidimension~$(p,p)$, there is a unique
decomposition of $T$ as a (possibly finite) weakly convergent series
$$T=\sum_{j\ge 1}\lambda_j[A_j]+R,~~~~\lambda_j>0,$$
where $[A_j]$ is the current of integration over an irreducible
$p$-dimensional ana\-lytic set $A_j\subset X$ and where $R$ is a closed
positive current with the property that $\dim E_c(R)<p$ for
every~$c>0$.
\endclaim

\proof{Proof of uniqueness.} If $T$ has such a decomposition, the
$p$-dimensional components of $E_c(T)$ are $(A_j)_{\lambda_j\ge c}$,
for $\nu(T,x)=\sum\lambda_j\nu([A_j],x)+\nu(R,x)$ is non zero
only on $\bigcup A_j\cup\bigcup E_c(R)$, and is equal to
$\lambda_j$ generically on $A_j$ $\big($more precisely,
$\nu(T,x)=\lambda_j$ at every regular
point of $A_j$ which does not belong to any intersection $A_j\cup A_k$,
$k\ne j$ or to $\bigcup E_c(R)\big)$. In particular $A_j$ and
$\lambda_j$ are unique.
\endproof

\proof{Proof of existence.} Let $(A_j)_{j\ge 1}$ be the countable collection of
$p$-dimensional components occurring in one of the sets $E_c(T)$,
$c\in\bQ_+^\star$, and let $\lambda_j>0$ be the generic Lelong
number of $T$ along $A_j$. Then Prop.~8.16 shows by induction on $N$
that $R_N=T-\sum_{1\le j\le N}\lambda_j[A_j]$ is positive. As $R_N$
is a decreasing sequence, there must be a
limit $R=\lim_{N\to+\infty}R_N$ in the weak topology. Thus we have
the asserted decomposition. By construction, $R$ has zero generic
Lelong number along $A_j$, so $\dim E_c(R)<p$ for every $c>0$.\qed
\endproof

It is very important to note that some components of lower dimension
can actually occur in $E_c(R)$, but they cannot be subtracted because
$R$ has bidimension $(p,p)$. A typical case is the case of a
bidimension \hbox{$(n-1,n-1)$} current $T=dd^cu$ with
$u=\log(|F_j|^{\gamma_1}+\ldots|F_N|^{\gamma_N})$ and
$F_j\in\cO(X)$. In general $\bigcup E_c(T)=\bigcap F_j^{-1}(0)$ has
dimension~$<n-1$. In that case, an important formula due to King
plays the role of (8.17). We state it in a somewhat more general form
than its original version \cite{King 1970}.

\claim{(8.18) King's formula} Let $F_1,\ldots,F_N$ be holomorphic
functions on a complex manifold~$X$, such that the zero variety
$Z=\bigcap F_j^{-1}(0)$ has codimension~$\ge p$, and set
$u=\log\sum|F_j|^{\gamma_j}$ with arbitrary coefficients $\gamma_j>0$.
Let $(Z_k)_{k\ge1}$ be the irreducible components of $Z$ of codimension $p$
exactly. Then there exist multiplicities $\lambda_k>0$ such that
$$(dd^cu)^p=\sum_{k\ge 1}\lambda_k[Z_k]+R,$$
where $R$ is a closed positive current such that $\bOne_ZR=0$
and $\codim E_c(R)>p$ for every $c>0$. Moreover the multiplicities
$\lambda_k$ are integers if $\gamma_1,\ldots,\gamma_N$ are integers,
and $\lambda_k=\gamma_1\ldots\gamma_p$ if $\gamma_1\le\ldots\le\gamma_N$
and some partial Jacobian determinant of $(F_1,\ldots,F_p)$ of order $p$ does
not vanish identically along~$Z_k$.
\endclaim

\proof{Proof} Observe that $(dd^cu)^p$ is well defined thanks to Cor.~4.11.
The comparison theorem~7.8 applied with $\varphi(z)=\log|z-x|$,
$v_1=\ldots=v_p=u$, $u_1=\ldots=u_p=\varphi$ and $T=1$ shows that the
Lelong number of $(dd^cu)^p$ is equal to $0$ at every point of~$X\ssm Z$.
Hence $E_c((dd^cu)^p)$ is contained in $Z$ and its $(n-p)$-dimensional
components are members of the family $(Z_k)$. The asserted decomposition
follows from Siu's formula~8.16. We must have
$\bOne_{Z_k}R=0$ for all irreducible components of~$Z$: when
$\codim Z_k>p$ this is automatically true, and when $\codim Z_k=p$ this
follows from (8.16) and the fact that $\codim E_c(R)>p$. If
$\det(\partial F_j/\partial z_k)_{1\le j,k\le p}\ne 0$ at some point
$x_0\in Z_k$, then $(Z,x_0)=(Z_k,x_0)$ is a smooth germ defined by the
equations $F_1=\ldots=F_p=0$. If we denote
$v=\log\sum_{j\le p}|F_j|^{\gamma_j}$ with $\gamma_1\le\ldots\le\gamma_N$,
then $u\sim v$ near $Z_k$ and
Th.~7.8 implies $\nu((dd^cu)^p,x)=\nu((dd^cv)^p,x)$ for all $x\in Z_k$
near $x_0$. On the other hand, if $G:=(F_1,\ldots,F_p):X\to\bC^p$, Cor.~7.4 gives 
$$(dd^cv)^p=G^\star\Big(dd^c\log\sum_{1\le j\le p}|z_j|^{\gamma_j}\Big)^p=
\gamma_1\ldots\gamma_p\,G^\star\delta_0=\gamma_1\ldots\gamma_p\,[Z_k]$$
near~$x_0$. This implies that the generic Lelong number of $(dd^cu)^p$
along $Z_k$ is $\lambda_k=\gamma_1\ldots\gamma_p$. The integrality of
$\lambda_k$ when $\gamma_1,\ldots,\gamma_N$ are integers will be proved in
the next section.\qed
\endproof

\section{9}{Transformation of Lelong Numbers by Direct Images}
Let $F:X\to Y$ be a holomorphic map between complex manifolds of respective
dimensions $\dim X=n$, $\dim Y=m$, and let $T$ be a closed positive current
of bidimension $(p,p)$ on~$X$. If $F_{\restriction\Supp\,T}$ is proper, the
direct image $F_\star T$ is defined by
$$\langle F_\star T,\alpha\rangle=\langle T,F^\star\alpha\rangle
\leqno(9.1)$$
for every test form $\alpha$ of bidegree $(p,p)$ on $Y$. This makes sense
because $\Supp\,T\cap F^{-1}(\Supp\,\alpha)$ is compact. It is easily seen
that $F_\star T$
is a closed positive current of bidimension $(p,p)$ on $Y$. 

\claim{(9.2) Example} \rm Let $T=[A]$ where $A$ is a $p$-dimensional irreducible analytic
set in $X$ such that $F_{\restriction A}$ is proper. We know by Remmert's
theorem~2.7.8 that $F(A)$ is an analytic set in~$Y$. Two cases may occur.
Either $F_{\restriction A}$ is generically finite and $F$ induces
an \'etale covering $A\ssm F^{-1}(Z)\longrightarrow F(A)\ssm Z$ for some nowhere dense
analytic subset $Z\subset F(A)$, or $F_{\restriction A}$ has generic fibers
of positive dimension and $\dim F(A)<\dim A$. In the first case, let
$s<+\infty$ be the covering degree. Then for every test form $\alpha$ of
bidegree $(p,p)$ on $Y$ we get
$$\langle F_\star[A],\alpha\rangle=\int_A F^\star\alpha=\int_{A\ssm F^{-1}(Z)}
F^\star\alpha=s\int_{F(A)\ssm Z}\alpha=s\,\langle[F(A)],\alpha\rangle$$
because $Z$ and $F^{-1}(Z)$ are negligible sets. Hence $F_\star[A]=
s[F(A)]$. On the other hand, if $\dim F(A)<\dim A=p$, the restriction
of $\alpha$ to $F(A)_\reg$ is zero, and therefore so is this the
restriction of $F^\star\alpha$ to $A_\reg$. Hence $F_\star[A]=0$.\qed
\endclaim

Now, let $\psi$ be a continuous plurisubharmonic function on $Y$ which is
semi-exhaustive on $F(\Supp\,T)$ (this set certainly contains
$\Supp F_\star T$). Since $F_{\restriction\Supp\,T}$ is proper, it follows
that $\psi\circ F$ is semi-exhaustive on $\Supp\,T$, for
$$\Supp\,T\cap\{\psi\circ F<R\}=F^{-1}\big(F(\Supp\,T)\cap\{\psi<R\}\big).$$

\claim{(9.3) Proposition} If $F(\Supp\,T)\cap\{\psi<R\}\compact Y$,
we have
$$\nu(F_\star T,\psi,r)=\nu(T,\psi\circ F,r)~~~~\hbox{\rm for all}~r<R,$$
in particular $\nu(F_\star T,\psi)=\nu(T,\psi\circ F)$.
\endclaim

Here, we do not necessarily assume that $X$ or $Y$ are Stein; we thus
replace $\psi$ with $\psi_{\gge s}=\max\{\psi,s\}$, $s<r$, in the definition of
$\nu(F_\star T,\psi,r)$ and $\nu(T,\psi\circ F,r)$.

\proof{Proof} The first equality can be written
$$\int_Y F_\star T\wedge\bOne_{\{\psi<r\}}(dd^c\psi_{\gge s})^p=
\int_X T\wedge(\bOne_{\{\psi<r\}}\circ F)(dd^c\psi_{\gge s}\circ F)^p.$$
This follows almost immediately from the adjunction formula (9.1) when
$\psi$ is smooth and when we write $\bOne_{\{\psi<R\}}=\lim\uparrow
g_k$ for some sequence of smooth functions $g_k$. In general, we write
$\psi_{\gge s}$ as a decreasing limit of smooth plurisubharmonic
functions and we apply our monotone continuity theorems (if~$Y$ is not Stein,
Richberg's theorem shows that we can obtain a decreasing sequence of almost
plurisubharmonic approximations such that the negative part of $dd^c$ converges
uniformly to~$0\,$; this is good enough to apply the monotone continuity
theorem; note that the integration is made on compact subsets, thanks to the
semi-exhaustivity assumption on $\psi$).\qed
\endproof

It follows from this that understanding the transformation of Lelong numbers
under direct images is equivalent to understanding the effect of $F$ on the
weight. We are mostly interested in computing the ordinary Lelong numbers
$\nu(F_\star T,y)$ associated with the weight $\psi(w)=\log|w-y|$ in some
local coordinates $(w_1,\ldots,w_m)$ on $Y$ near~$y$. Then Prop.~9.3 gives
$$\leqalignno{
\nu(F_\star T,y)&=\nu(T,\log|F-y|)~~~~\hbox{\rm with}&(9.4)\cr
\log|F(z)-y|&={1\over 2}\log\sum|F_j(z)-y_j|^2,~~~~F_j=w_j\circ F.\cr}$$
We are going to show that $\nu(T,\log|F-y|)$ is bounded below by a linear
combination of the Lelong numbers of $T$ at points $x$ in the fiber
$F^{-1}(y)$, with suitable multiplicities attached to $F$ at these points.
These multiplicities can be seen as generalizations of the notion of
multiplicity of an analytic map introduced by \cite{Stoll 1966}.

\claim{(9.5) Definition} Let $x\in X$ and $y=F(x)$. Suppose that the
codimension of the fiber $F^{-1}(y)$ at $x$ is $\ge p$. Then we set
$$\mu_p(F,x)=\nu\big((dd^c\log|F-y|)^p,x\big).$$
\endclaim

Observe that $(dd^c\log|F-y|)^p$ is well defined thanks to Cor.~4.10.
The second comparison theorem~7.8 immediately shows that $\mu_p(F,x)$
is independent of the choice of local coordinates on~$Y$ (and also on $X$,
since Lelong nombers do not depend on coordinates). By definition,
$\mu_p(F,x)$ is the mass carried by $\{x\}$ of the measure
$$(dd^c\log|F(z)-y|)^p\wedge(dd^c\log|z-x|)^{n-p}.$$
We are going to give a more geometric interpretation of this
multiplicity, from which it will follow that $\mu_p(F,x)$ is always a
positive integer (in particular, the proof of (8.18) will be complete).

\claim{(9.6) Example} \rm For $p=n=\dim X$, the assumption $\codim_x F^{-1}(y)\ge p$
means that the germ of map $F:(X,x)\longrightarrow(Y,y)$ is finite. Let $U_x$ be a
neighborhood of $x$ such that $\ol U_x\cap F^{-1}(y)=\{x\}$, let $W_y$
be a neighborhood of $y$ disjoint from $F(\partial U_x)$ and let
$V_x=U_x\cap F^{-1}(W_y)$. Then $F:V_x\to W_y$ is proper and finite,
and we have $F_\star[V_x]=s\,[F(V_x)]$ where $s$ is the local covering
degree of $F:V_x\to F(V_x)$ at~$x$. Therefore
$$\eqalign{
\mu_n(F,x)&=\int_{\{x\}}\big(dd^c\log|F-y|\big)^n=\nu\big([V_x],\log|F-y|\big)
=\nu\big(F_\star[V_x],y\big)\cr
&=s\,\nu\big(F(V_x),y\big).\cr}$$
In the particular case when $\dim Y=\dim X$, we have $(F(V_x),y)=(Y,y)$, so
$\mu_n(F,x)=s$. In general, it is a well known fact that the ideal
generated by $(F_1-y_1,\ldots,F_m-y_m)$ in $\cO_{X,x}$ has the same integral
closure as the ideal generated by $n$ generic linear combinations of the
generators, that is, for a generic choice of coordinates $w'=(w_1,\ldots,w_n)$,
$w''=(w_{n+1},\ldots,w_m)$ on $(Y,y)$, we have $|F(z)-y|\le C|w'\circ F(z)|$
(this is a simple consequence of Lemma~7.5 applied to $A=F(V_x)$).
Hence for $p=n$, the comparison theorem~7.1 gives
$$\mu_n(F,x)=\mu_n(w'\circ F,x)=\hbox{\rm local covering degree of}~
w'\circ F~~{\rm at}~x,$$
for a generic choice of coordinates $(w',w'')$ on $(Y,y)$.\qed
\endclaim

\noindent{\bf (9.7) Geometric interpretation of $\mu_p(F,x)$.} An
application of Crofton's formula 7.11 shows, after a translation, that
there is a small ball $B(x,r_0)$ on which
$$\leqalignno{
(dd^c\log|F(z)-y|)^p\wedge(dd^c\log|z-x|)^{n-p}&{}=\cr
\int_{S\in G(p,n)}
(dd^c\log|F(z)-y|)^p&{}\wedge[x+S]\,dv(S).&(9.7\,{\rm a})\cr}$$
For a rigorous proof of (9.7$\,$a), we replace $\log|F(z)-y|$ by the
``regularized''
function ${1\over2}\log(|F(z)-y|^2+\varepsilon^2)$ and let $\varepsilon$
tend to~$0$ on both sides. By~(4.3) (resp.\ by (4.10)), the wedge product
$(dd^c\log|F(z)-y|)^p\wedge[x+S]$ is well defined on a small ball $B(x,r_0)$
as soon as $x+S$ does not intersect \hbox{$F^{-1}(y)\cap\partial B(x,r_0)$}
(resp.\ intersects $F^{-1}(y)\cap B(x,r_0)$ at finitely many points);
thanks to the assumption $\codim(F^{-1}(y),x)\ge p$, Sard's theorem shows
that this is the case for all $S$ outside a negligible closed subset $E$
in~$G(p,n)$ (resp.\ by Bertini, an analytic subset $A$ in $G(p,n)$ with
$A\subset E$). Fatou's lemma then implies that the inequality
$\ge$ holds in (9.7$\,$a). To get equality, we observe that we have
bounded convergence on all complements $G(p,n)\ssm V(E)$ of
neighborhoods $V(E)$ of~$E$. However the mass of
$\int_{V(E)}[x+S]\,dv(S)$ in $B(x,r_0)$ is proportional to
$v(V(E))$ and therefore tends to $0$ when $V(E)$ is small;
this is sufficient to complete the proof, since Prop.~4.6~b) gives
$$\int_{z\in\ol B(x,r_0)}\big(dd^c\log(|F(z)-y|^2+\varepsilon^2)\big)^p
\wedge\int_{S\in V(E)}[x+S]\,dv(S)\le C\,v(V(E))$$
with a constant $C$ independent of~$\varepsilon$. By evaluating
(9.7$\,$a) on $\{x\}$, we get
$$\mu_p(F,x)=\int_{S\in G(p,n)\ssm A}
\nu\big((dd^c\log|F_{\restriction x+S}-z|)^p,x\big)\,dv(S).
\leqno(9.7\,{\rm b})$$
Let us choose a linear parametrization $g_S:\bC^p\to S$ depending
analytically on local coordinates of $S$ in~$G(p,n)$. Then
Theorem~8.4 with $T=[\bC^p]$ and $\varphi(z,S)=\log|F\circ g_S(z)-y|$
shows that 
$$\nu\big((dd^c\log|F_{\restriction x+S}-z|)^p,x\big)=
\nu\big([\bC^p],\log|F\circ g_S(z)-y|\big)$$
is Zariski upper semicontinuous in $S$ on $G(p,n)\ssm A$. However,
(9.6) shows that these numbers are integers, so
$S\mapsto\nu\big((dd^c\log|F_{\restriction x+S}-z|)^p,x\big)$ must be
constant on a Zariski open subset in~$G(p,n)$. By (9.7$\,$b), we obtain
$$\mu_p(F,x)=\mu_p(F_{\restriction x+S},x)=\hbox{\rm local degree of}~
w'\circ F_{\restriction x+S}~~\hbox{\rm at}~x\leqno(9.7\,{\rm c})$$
for generic subspaces $S\in G(p,n)$ and generic coordinates
$w'=(w_1,\ldots,w_p)$, $w''=(w_{p+1},\ldots,w_m)$ on~$(Y,y)$.\qed

\claim{(9.8) Example} \rm Let $F:\bC^n\longrightarrow\bC^n$ be defined by
$$F(z_1,\ldots,z_n)=(z_1^{s_1},\ldots,z_n^{s_n}),~~~~s_1\le\ldots\le s_n.$$
We claim that $\mu_p(F,0)=s_1\ldots s_p$. In fact, for a generic
$p$-dimensional subspace $S\subset\bC^n$ such that $z_1,\ldots,z_p$
are coordinates on $S$ and $z_{p+1},\ldots,z_n$ are linear forms in
$z_1,\ldots,z_p$, and for generic coordinates $w'=(w_1,\ldots,w_p)$,
$w''=(w_{p+1},\ldots,w_n)$ on $\bC^n$, we can
rearrange $w'$ by linear combinations so that
$w_j\circ F_{\restriction S}$ is a linear combination of
$(z_j^{s_j},\ldots,z_n^{s_n})$ and has non zero coefficient in $z_j^{s_j}$
as a polynomial in $(z_j,\ldots,z_p)$.
It is then an exercise to show that $w'\circ F_{\restriction S}$ has
covering degree $s_1\ldots s_p$ at~$0$ [$\,$compute inductively
the roots $z_n$, $z_{n-1},\ldots,z_j$ of $w_j\circ F_{\restriction S}(z)=a_j$
and use Lemma~II.3.10 to show that the $s_j$ values of $z_j$
lie near~$0$ when $(a_1,\ldots,a_p)$ are small$\,$].\qed
\endclaim

We are now ready to prove the main result of this section, which des\-cribes
the behaviour of Lelong numbers under proper morphisms. A~similar weaker
result was already proved in \cite{Demailly 1982b} with some other non optimal
multiplicities $\mu_p(F,x)$.

\claim{(9.9) Theorem} Let $T$ be a closed positive current of bidimension
$(p,p)$ on $X$ and let $F:X\longrightarrow Y$ be an analytic map such that the
restriction $F_{\restriction\Supp\,T}$ is proper. Let $I(y)$ be the
set of points $x\in\Supp\,T\cap F^{-1}(y)$ such that $x$ is equal to
its connected component in $\Supp\,T\cap F^{-1}(y)$ and
$\codim(F^{-1}(y),x)\ge p$. Then we have
$$\nu(F_\star T,y)\ge\sum_{x\in I(y)}\mu_p(F,x)\,\nu(T,x).$$
\endclaim

In particular, we have $\nu(F_\star T,y)\ge\sum_{x\in I(y)}\nu(T,x)$.
This inequality no longer holds if the summation is extended to
all points $x\in\Supp\,T\cap F^{-1}(Y)$ and if this set contains
positive dimensional connected components: for example, if $F:X\longrightarrow Y$
contracts some exceptional subspace $E$ in $X$ to a point~$y_0$
(e.g.\ if $F$ is a blow-up map, see \S~7.12), then $T=[E]$ has direct
image $F_\star[E]=0$ thanks to~(9.2).

\proof{Proof} We proceed in three steps.
\smallskip
\noindent{\it Step 1. Reduction to the case of a single point $x$ in the
fiber.} It is sufficient to prove the inequality when the summation is
taken over an arbitrary finite subset $\{x_1,\ldots,x_N\}$ of~$I(y)$.
As $x_j$ is equal to its connected component in $\Supp\,T\cap F^{-1}(y)$,
it has a fondamental system of relative open-closed
neighborhoods, hence there are disjoint neighborhoods $U_j$ of $x_j$
such that $\partial U_j$ does not intersect $\Supp\,T\cap F^{-1}(y)$. Then
the image $F(\partial U_j\cap\Supp\,T)$ is a closed set
which does not contain~$y$. Let $W$ be a neighborhood of $y$ disjoint from
all sets $F(\partial U_j\cap\Supp\,T)$, and let $V_j=U_j\cap F^{-1}(W)$.
It is clear that $V_j$ is a neighborhood of $x_j$ and that
$F_{\restriction V_j}:V_j\to W$ has a proper
restriction to $\Supp\,T\cap V_j$. Moreover, we obviously have
$F_\star T\ge \sum_j (F_{\restriction V_j})_\star T$ on~$W$.
Therefore, it is enough to check the inequality
$\nu(F_\star T,y)\ge\mu_p(F,x)\,\nu(T,x)$ for a single point $x\in I(y)$,
in the case when $X\subset\bC^n$, $Y\subset\bC^m$ are open
subsets and~$x=y=0$.
\smallskip
\noindent{\it Step 2. Reduction to the case when $F$ is finite.}
By (9.4), we have
$$\eqalign{
\nu(F_\star T,0)&=\inf_{V\ni 0}\int_V T\wedge(dd^c\log|F|)^p\cr
&=\inf_{V\ni 0}\lim_{\varepsilon\to 0}\int_V T\wedge
\big(dd^c\log(|F|+\varepsilon|z|^N)\big)^p,\cr}$$
and the integrals are well defined as soon as $\partial V$ does
not intersect the set $\Supp\,T\cap F^{-1}(0)$ (may be after replacing
$\log|F|$ by $\max\{\log|F|,s\}$ with $s\ll 0$). For every $V$ and
$\varepsilon$, the last integral is larger than $\nu(G_\star T,0)$ where
$G$ is the finite morphism defined by
$$G:X\longrightarrow Y\times\bC^n,~~~~(z_1,\ldots,z_n)\longmapsto
(F_1(z),\ldots,F_m(z),z_1^N,\ldots,z_n^N).$$
We claim that for $N$ large enough we have $\mu_p(F,0)=\mu_p(G,0)$.
In fact, $x\in I(y)$ implies by definition $\codim(F^{-1}(0),0)\ge p$.
Hence, if $S=\{u_1=\ldots=u_{n-p}=0\}$ is a generic $p$-dimensional
subspace of $\bC^n$, the germ of variety $F^{-1}(0)\cap S$ defined by
$(F_1,\ldots,F_m,u_1,\ldots,u_{n-p})$ is~$\{0\}$. Hilbert's Nullstellensatz
implies that some powers of $z_1,\ldots,z_n$ are in the ideal $(F_j,u_k)$.
Therefore $|F(z)|+|u(z)|\ge C|z|^a$ near $0$ for some integer $a$ independent
of $S$ (to see this, take coefficients of the $u_k$'s as additional variables);
in particular $|F(z)|\ge C|z|^a$ for $z\in S$ near~$0$. The comparison
theorem~7.1 then shows that $\mu_p(F,0)=\mu_p(G,0)$ for $N\ge a$.
If we are able to prove that $\nu(G_\star T,0)\ge\mu_p(G,0)\nu(T,0)$ in
case $G$ is finite, the obvious inequality
$\nu(F_\star T,0)\ge\nu(G_\star T,0)$ concludes the proof.

\noindent{\it Step 3. Proof of the inequality
$\nu(F_\star T,y)\ge\mu_p(F,x)\,\nu(T,x)$ when $F$ is finite and
$F^{-1}(y)=x$.} Then $\varphi(z)=\log|F(z)-y|$ has a single isolated
pole at $x$ and we have $\mu_p(F,x)=\nu((dd^c\varphi)^p,x)$. It is
therefore sufficient to apply to following Proposition.
\endproof

\claim{(9.10) Proposition} Let $\varphi$ be a semi-exhaustive continuous
plurisubharmonic function on $X$ with a single isolated pole at~$x$. Then
$$\nu(T,\varphi)\ge\nu(T,x)\,\nu((dd^c\varphi)^p,x).$$
\endclaim

\proof{Proof} Since the question is local, we can suppose that $X$ is the ball
$B(0,r_0)$ in $\bC^n$ and $x=0$. Set $X'=B(0,r_1)$ with $r_1<r_0$
and $\Phi(z,g)=\varphi\circ g(z)$ for \hbox{$g\in{\rm Gl}_n(\bC)$.} Then
there is a small neighborhood $\Omega$ of the unitary group
\hbox{$U(n)\subset{\rm Gl}_n(\bC)$} such that $\Phi$ is
plurisubharmonic on $X'\times\Omega$ and semi-exhaustive with respect to~$X'$.
Theorem~8.4 implies that the map $g\mapsto\nu(T,\varphi\circ g)$ is Zariski
upper semi-continuous on~$\Omega$. In particular, we must have
$\nu(T,\varphi\circ g)\le\nu(T,\varphi)$ for all $g\in\Omega\ssm A$ in the
complement of a complex analytic set~$A$. Since ${\rm Gl}_n(\bC)$ is the
complexification of~$U(n)$, the intersection $U(n)\cap A$ must be a nowhere
dense real analytic subset of~$U(n)$. Therefore, if $dv$ is the Haar measure
of mass $1$ on~$U(n)$, we have
$$\leqalignno{
\nu(T,\varphi)&\ge\int_{g\in U(n)}\nu(T,\varphi\circ g)\,dv(g)\cr
&=\lim_{r\to 0}\int_{g\in U(n)}dv(g)
\int_{B(0,r)}T\wedge(dd^c\varphi\circ g)^p.&(9.11)\cr}$$
Since $\int_{g\in U(n)}(dd^c\varphi\circ g)^pdv(g)$ is a unitary
invariant $(p,p)$-form on~$B$, Lemma 7.10 implies
$$\int_{g\in U(n)}(dd^c\varphi\circ g)^pdv(g)=\big(dd^c\chi(\log|z|)\big)^p$$
where $\chi$ is a convex increasing function. The Lelong number at $0$ of the
left hand side is equal to $\nu((dd^c\varphi)^p,0)$, and must be equal to the
Lelong number of the right hand side, which is $\lim_{t\to-\infty}\chi'(t)^p$
(to see this, use either Formula~(5.5) or Th.~7.8). Thanks to the last
equality, Formulas (9.11) and (5.5) imply
$$\eqalignno{
\nu(T,\varphi)&\ge\lim_{r\to 0}
\int_{B(0,r)}T\wedge\big(dd^c\chi(\log|z|)\big)^p\cr
&=\lim_{r\to 0}\chi'(\log r-0)^p\nu(T,0,r)
\ge\nu((dd^c\varphi)^p,0)\,\nu(T,0).&\square\cr}$$
\endproof

Another interesting question is to know whether it is possible to get
inequalities in the opposite direction, i.e.\ to find upper bounds for
$\nu(F_\star T,y)$ in terms of the Lelong numbers $\nu(T,x)$.
The example $T=[\Gamma]$ with the curve $\Gamma:
t\mapsto(t^a,t^{a+1},t)$ in $\bC^3$ and $F:\bC^3\to\bC^2$,
$(z_1,z_2,z_3)\mapsto(z_1,z_2)$, for which $\nu(T,0)=1$ and
$\nu(F_\star T,0)=a$, shows that this may be possible only when $F$
is finite. In this case, we have:

\claim{(9.12) Theorem} Let $F:X\to Y$ be a proper and finite analytic map and let
$T$ be a closed positive current of bidimension $(p,p)$ on~$X$. Then
$$\nu(F_\star T,y)\le\sum_{x\in\Supp\,T\cap F^{-1}(y)}\ol\mu_p(F,x)\,
\nu(T,x)\leqno({\rm a})$$
where $\ol\mu_p(F,x)$ is the multiplicity defined as follows: if
$H:(X,x)\to(\bC^n,0)$ is a germ of finite map, we set
$$\leqalignno{
\sigma(H,x)&=\inf\big\{\alpha>0\,;\,\exists C>0,\,|H(z)|\ge C|z-x|^\alpha~
\hbox{\rm near}~x\big\},&({\rm b})\cr
\ol\mu_p(F,x)&=\inf_G{\sigma(G\circ F,x)^p\over\mu_p(G,0)},&({\rm c})\cr}$$
where $G$ runs over all germs of maps $(Y,y)\longrightarrow(\bC^n,0)$ such that
$G\circ F$ is finite.
\endclaim

\proof{Proof} If $F^{-1}(y)=\{x_1,\ldots,x_N\}$, there is a neighborhood $W$ of $y$
and disjoint neighborhoods $V_j$ of $x_j$ such that $F^{-1}(W)=\bigcup V_j$.
Then $F_\star T=\sum(F_{\restriction V_j})_\star T$ on $W$, so it is
enough to consider the case when $F^{-1}(y)$ consists of a single point~$x$.
Therefore, we assume that $F:V\to W$ is proper and finite, where
$V$, $W$ are neighborhoods of $0$ in $\bC^n$, $\bC^m$ and $F^{-1}(0)=\{0\}$.
Let \hbox{$G:(\bC^m,0)\longrightarrow(\bC^n,0)$} be a germ of map such that $G\circ F$
is finite. Hilbert's Nullstellensatz shows that there exists $\alpha>0$ and
$C>0$ such that $|G\circ F(z)|\ge C|z|^\alpha$ near~$0$. Then the comparison
theorem~7.1 implies
$$\nu(G_\star F_\star T,0)=\nu(T,\log|G\circ F|)\le\alpha^p\nu(T,\log|z|)=
\alpha^p\nu(T,0).$$
On the other hand, Th.~9.9 applied to $\Theta=F_\star T$ on $W$ gives
$$\nu(G_\star F_\star T,0)\ge\mu_p(G,0)\,\nu(F_\star T,0).$$
Therefore
$$\nu(F_\star T,0)\le{\alpha^p\over\mu_p(G,0)}\nu(T,0).$$
The infimum of all possible values of $\alpha$ is by definition
$\sigma(G\circ F,0)$, thus by taking the infimum over $G$ we obtain
$$\nu(F_\star T,0)\le\ol\mu_p(F,0)\,\nu(T,0).\eqno{\square}$$
\endproof

\claim{(9.13) Example} \rm Let $F(z_1,\ldots,z_n)=(z_1^{s_1},\ldots,z_n^{s_n})$,
$s_1\le\ldots\le s_n$ as in~9.8. Then we have
$$\mu_p(F,0)=s_1\ldots s_p,~~~~~~\ol\mu_p(F,0)=s_{n-p+1}\ldots s_n.$$
To see this, let $s$ be the lowest common multiple of $s_1,\ldots,s_n$
and let $G(z_1,\ldots,z_n)=(z_1^{s/s_1},\ldots,z_n^{s/s_n})$. Clearly
$\mu_p(G,0)=(s/s_{n-p+1})\ldots(s/s_n)$ and $\sigma(G\circ F,0)=s$, so
we get by definition $\ol\mu_p(F,0)\le s_{n-p+1}\ldots s_n$.
Finally, if $T=[A]$ is the current of integration over the $p$-dimensional
subspace $A=\{z_1=\ldots=z_{n-p}=0\}$, then $F_\star[A]=
s_{n-p+1}\ldots s_n\,[A]$ because $F_{\restriction A}$
has covering degree $s_{n-p+1}\ldots s_n$. Theorem~9.12 shows that
we must have $s_{n-p+1}\ldots s_n\le\ol\mu_p(F,0)$, QED.
If $\lambda_1\le\ldots\le\lambda_n$ are positive real numbers and
$s_j$ is taken to be the integer part of $k\lambda_j$ as $k$
tends to $+\infty$, Theorems~9.9 and 9.12 imply in the limit
the following:
\endclaim

\claim{(9.14) Corollary} For $0<\lambda_1\le\ldots\le\lambda_n$,
Kiselman's directional Lelong numbers satisfy the inequalities
$$\lambda_1\ldots\lambda_p\,\nu(T,x)\le\nu(T,x,\lambda)\le
\lambda_{n-p+1}\ldots\lambda_n\,\nu(T,x).\eqno{\square}$$
\endclaim

\claim{(9.15) Remark} \rm It would be interesting to have a direct geometric
interpretation of~$\ol\mu_p(F,x)$. In fact, we do not even know whether
$\ol\mu_p(F,x)$ is always an integer.
\endclaim

\section{10}{A Schwarz Lemma. Application to Number Theory}
In this section, we show how Jensen's formula and Lelong numbers can be
used to prove a fairly general Schwarz lemma relating growth and zeros
of entire functions in~$\bC^n$.
In order to simplify notations, we denote by $|F|_r$ the supremum of
the modulus of a function $F$ on the ball of center $0$ and radius~$r$.
Then, following \cite{Demailly~1982a}, we present some applications
with a more arithmetical flavour.

\claim{(10.1) Schwarz lemma} Let $P_1,\ldots,P_N\in\bC[z_1,\ldots,z_n]$ be
polynomials of degree~$\delta$, such that their homogeneous parts of
degree $\delta$ do not vanish simultaneously except at~$0$. Then there
is a constant \hbox{$C\ge 2$} such that for all entire functions
$F\in\cO(\bC^n)$ and all $R\ge r\ge 1$ we have
$$\log|F|_r\le\log|F|_R-\delta^{1-n}\nu([Z_F],\log|P|)\,\log{R\over Cr}$$
where $Z_F$ is the zero divisor of $F$ and $P=(P_1,\ldots,P_N):\bC^n\longrightarrow
\bC^N$. Moreover
$$\nu([Z_F],\log|P|)\ge\sum_{w\in P^{-1}(0)}\ord(F,w)\,\mu_{n-1}(P,w)$$
where $\ord(F,w)$ denotes the vanishing order of $F$ at $w$ and
$\mu_{n-1}(P,w)$ is the $(n-1)$-multiplicity of $P$ at $w$, as defined
in $(9.5)$ and $(9.7)$.
\endclaim

\proof{Proof} Our assumptions imply that $P$ is a proper and finite map. The last
inequality is then just a formal consequence
of formula (9.4) and Th.~9.9 applied to~$T=[Z_F]$. Let $Q_j$ be
the homogeneous part of degree $\delta$ in~$P_j$.
For~$z_0\in B(0,r)$, we introduce the weight functions
$$\varphi(z)=\log|P(z)|,~~~~~~\psi(z)=\log|Q(z-z_0)|.$$
Since $Q^{-1}(0)=\{0\}$ by hypothesis, the homogeneity of $Q$ shows
that there are constants $C_1,C_2>0$ such that
$$C_1|z|^\delta\le|Q(z)|\le C_2|z|^\delta~~~~\hbox{\rm on}~~\bC^n.
\leqno(10.2)$$
The homogeneity also implies $(dd^c\psi)^n=\delta^n\,\delta_{z_0}$.
We apply the Lelong Jensen formula~6.5 to the measures $\mu_{\psi,s}$
associated with $\psi$ and to~$V=\log|F|$. This gives
$$\mu_{\psi,s}(\log|F|)-\delta^n\log|F(z_0)|=\int_{-\infty}^sdt\int_{\{\psi<t\}}
[Z_F]\wedge(dd^c\psi)^{n-1}.\leqno(10.3)$$
By (6.2), $\mu_{\psi,s}$ has total mass $\delta^n$ and has support in
$$\{\psi(z)=s\}=\{Q(z-z_0)=e^s\}\subset B\big(0,r+(e^s/C_1)^{1/\delta}\big).$$
Note that the inequality in the Schwarz lemma is obvious if $R\le Cr$, so
we can assume $R\ge Cr\ge 2r$. We take $s=\delta\log(R/2)+\log C_1\,$; then
$$\{\psi(z)=s\}\subset B(0,r+R/2)\subset B(0,R).$$
In particular, we get $\mu_{\psi,s}(\log|F|)\le \delta^n\log|F|_R$ and
formula~(10.3) gives
$$\log|F|_R-\log|F(z_0)|\ge\delta^{-n}\int_{s_0}^sdt\int_{\{\psi<t\}}
[Z_F]\wedge(dd^c\psi)^{n-1}\leqno(10.4)$$
for any real number $s_0<s$. The proof will be complete if we are able to
compare the integral in (10.4) to the corresponding integral with $\varphi$
in place of~$\psi$. The argument for this is quite similar to the proof
of the comparison theorem, if we observe that $\psi\sim\varphi$ at
infinity. We introduce the auxiliary function
$$w=\cases{
\max\{\psi,(1-\varepsilon)\varphi+\varepsilon t-\varepsilon\}
&on $\{\psi\ge t-2\}$,\cr
(1-\varepsilon)\varphi+\varepsilon t-\varepsilon
&on $\{\psi\le t-2\}$,\cr}$$
with a constant $\varepsilon$ to be determined later, such that
$(1-\varepsilon)\varphi+\varepsilon t-\varepsilon>\psi$ near $\{\psi=t-2\}$ and
$(1-\varepsilon)\varphi+\varepsilon t-\varepsilon<\psi$ near $\{\psi=t\}$.
Then Stokes' theorem implies
$$\leqalignno{
\int_{\{\psi<t\}}&[Z_F]\wedge(dd^c\psi)^{n-1}=
\int_{\{\psi<t\}}[Z_F]\wedge(dd^cw)^{n-1}\cr
&\ge(1-\varepsilon)^{n-1}\int_{\{\psi<t-2\}}[Z_F]\wedge(dd^c\varphi)^{n-1}
\ge(1-\varepsilon)^{n-1}\nu([Z_F],\log|P|).&(10.5)\cr}$$
By (10.2) and our hypothesis $|z_0|<r$, the condition $\psi(z)=t$ implies
$$\eqalign{
|Q(z-z_0)|=e^t~~&\Longrightarrow~~
e^{t/\delta}/C_1^{1/\delta}\le|z-z_0|\le e^{t/\delta}/C_2^{1/\delta},\cr
|P(z)-Q(z-z_0)|&\le C_3(1+|z_0|)(1+|z|+|z_0|)^{\delta-1}\le
C_4 r(r+e^{t/\delta})^{\delta-1},\cr
\Big|{P(z)\over Q(z-z_0)}-1\Big|&\le C_4 re^{-t/\delta}
(re^{-t/\delta}+1)^{\delta-1}\le 2^{\delta-1}C_4 re^{-t/\delta},\cr}$$
provided that $t\ge \delta\log r$. Hence for $\psi(z)=t\ge s_0\ge
\delta\log(2^\delta C_4r)$, we get
$$|\varphi(z)-\psi(z)|=\Big|\log{|P(z)|\over|Q(z-z_0)|}\Big|
\le C_5 re^{-t/\delta}.$$
Now, we have
$$\big[(1-\varepsilon)\varphi+\varepsilon t-\varepsilon\big]-\psi=
(1-\varepsilon)(\varphi-\psi)+\varepsilon(t-1-\psi),$$
so this difference is $<C_5re^{-t/\delta}-\varepsilon$ on $\{\psi=t\}$
and $>-C_5re^{(2-t)/\delta}+\varepsilon$ on $\{\psi=t-2\}$. Hence it is
sufficient to take $\varepsilon=C_5re^{(2-t)/\delta}$. This number has to be
$<1$, so we take $t\ge s_0\ge 2+\delta\log(C_5r)$. Moreover, (10.5) actually
holds only if $P^{-1}(0)\subset\{\psi<t-2\}$, so by (10.2) it is enough to take
$t\ge s_0\ge 2+\log(C_2(r+C_6)^\delta)$ where $C_6$ is such that
$P^{-1}(0)\subset\ol B(0,C_6)$. Finally, we see that we can choose
$$s=\delta\log R-C_7,~~~~~s_0=\delta\log r+C_8,$$
and inequalities (10.4), (10.5) together imply
$$\log|F|_R-\log|F(z_0)|\ge\delta^{-n}\Big(\int_{s_0}^s
(1-C_5re^{(2-t)/\delta})^{n-1}\,dt\Big)\nu([Z_F],\log|P|).$$
The integral is bounded below by
$$\int_{C_8}^{\delta\log(R/r)-C_7}
(1-C_9e^{-t/\delta})\,dt\ge\delta\log(R/Cr).$$
This concludes the proof, by taking the infimum when $z_0$ runs
over $B(0,r)$.\qed
\endproof

\claim{(10.6) Corollary} Let $S$ be a finite subset of $\bC^n$ and let $\delta$
be the minimal degree of algebraic hypersurfaces containing~$S$. Then
there is a constant $C\ge 2$ such that for all $F\in\cO(\bC^n)$ and all
$R\ge r\ge 1$ we have
$$\log|F|_r\le\log|F|_R-\ord(F,S){\delta+n(n-1)/2\over n!}
\log{R\over Cr}$$
where $\ord(F,S)=\min_{w\in S}\ord(F,w)$.
\endclaim

\proof{Proof} In view of Th.~10.1, we only have to select suitable polynomials
$P_1,\ldots,P_N$. The vector space $\bC[z_1,\ldots,z_n]_{<\delta}$ of polynomials
of degree $<\delta$ in $\bC^n$ has dimension
$$m(\delta)={\delta+n-1\choose n}=
{\delta(\delta+1)\ldots(\delta+n-1)\over n!}.$$
By definition of $\delta$, the linear forms
$$\bC[z_1,\ldots,z_n]_{<\delta}\longrightarrow\bC,~~~~P\longmapsto P(w),~~w\in S$$
vanish simultaneously only when~$P=0$. Hence we can find $m=m(\delta)$
points $w_1,\ldots,$\break
$w_m\in S$ such that the linear forms $P\mapsto P(w_j)$
define a basis of~$\bC[z_1,\ldots,z_n]_{<\delta}^\star$. This means that there is
a unique polynomial $P\in\bC[z_1,\ldots,z_n]_{<\delta}$ which takes given values
$P(w_j)$ for $1\le j\le m$. In~particular, for every multiindex $\alpha$,
$|\alpha|=\delta$, there is a unique polynomial \hbox{$R_\alpha\in
\bC[z_1,\ldots,z_n]_{<\delta}$} such that $R_\alpha(w_j)=w_j^\alpha$. Then the
polynomials $P_\alpha(z)=z^\alpha-R_\alpha(z)$ have degree~$\delta$,
vanish at all points $w_j$ and their homogeneous parts of maximum degree
$Q_\alpha(z)=z^\alpha$ do not vanish simultaneously except at~$0$.
We simply use the fact that $\mu_{n-1}(P,w_j)\ge 1$ to get
$$\nu([Z_F],\log|P|)\ge\sum_{w\in P^{-1}(0)}\ord(F,w)\ge
m(\delta)\,\ord(F,S).$$
Theorem 10.1 then gives the desired inequality, because $m(\delta)$ is a
polynomial with positive coefficients and with leading terms 
$${1\over n!}\big(\delta^n+n(n-1)/2\,\delta^{n-1}+\ldots\big).\eqno{\square}$$
\endproof

Let $S$ be a finite subset of~$\bC^n$. According to \cite{Waldschmidt 1976},
we introduce for every integer $t>0$ a number $\omega_t(S)$ equal to
the minimal degree of polynomials $P\in\bC[z_1,\ldots,z_n]$ which vanish
at order $\ge t$ at every point of~$S$. The obvious subadditivity
property
$$\omega_{t_1+t_2}(S)\le\omega_{t_1}(S)+\omega_{t_2}(S)$$
easily shows that
$$\Omega(S):=\inf_{t>0}{\omega_t(S)\over t}=
\lim_{t\to+\infty}{\omega_t(S)\over t}.$$
We call $\omega_1(S)$ the {\it degree} of $S$ (minimal degree of algebraic
hypersurfaces containing~$S$) and $\Omega(S)$ the {\it singular degree}
of~$S$. If we apply Cor.~10.6 to a polynomial $F$ vanishing at order
$t$ on $S$ and fix $r=1$, we get
$$\log|F|_R\ge t{\delta+n(n-1)/2\over n!}\log{R\over C}+\log|F|_1$$
with $\delta=\omega_1(S)$, in particular
$${\rm deg}\,F\ge t{\omega_1(S)+n(n-1)/2\over n!}.$$
The minimum of ${\rm deg}\,F$ over all such $F$ is by
definition~$\omega_t(S)$. If we divide by $t$ and take the infimum
over~$t$, we get the interesting inequality
$${\omega_t(S)\over t}\ge\Omega(S)\ge{\omega_1(S)+n(n-1)/2\over n!}.
\leqno(10.7)$$

\claim{(10.8) Remark} \rm The constant ${\omega_1(S)+n(n-1)/2\over n!}$ 
in (10.6) and
(10.7) is optimal for $n=1,2$ but not for~$n\ge3$. It can be shown by means
of H\"ormander's $L^2$ estimates \cite{Waldschmidt 1978} that for every
$\varepsilon>0$
the Schwarz lemma (10.6) holds with coefficient $\Omega(S)-\varepsilon\,$:
$$\log|F|_r\le\log|F|_R-\ord(F,S)(\Omega(S)-\varepsilon)\log{R\over
C_\varepsilon r},$$
and that $\Omega(S)\ge(\omega_u(S)+1)/(u+n-1)$ for every $u\ge 1\,$;
this last inequality is due to \cite{Esnault-Viehweg 1983}, who used
deep tools of algebraic geometry; \cite{Azhari 1990} reproved it recently by
means of H\"ormander's $L^2$ estimates. Rather simple examples
\cite{Demailly 1982a} lead to the conjecture
$$\Omega(S)\ge{\omega_u(S)+n-1\over u+n-1}~~~~\hbox{for every}~~u\ge 1.$$
The special case $u=1$ of the conjecture was first stated by
\cite{Chudnovsky 1979}.
\endclaim

Finally, let us mention that Cor.~10.6 contains Bombieri's theorem
on algebraic values of meromorphic maps satisfying algebraic
differential equations \cite{Bombieri 1970}.
Recall that an entire function $F\in\cO(\bC^n)$ is said to be of
order $\le\rho$ if for every $\varepsilon>0$ there
is a constant $C_\varepsilon$ such that $|F(z)|\le C_\varepsilon\exp(
|z|^{\rho+\varepsilon})$. A meromorphic function is said to be of order
$\le\rho$ if it can be written $G/H$ where $G$, $H$ are entire functions
of order $\le\rho$. 

\claim{(10.9) Theorem {\rm(\cite{Bombieri 1970})}} Let $F_1,\ldots,F_N$
be meromorphic functions on $\bC^n$, such that $F_1,\ldots,F_d$,
$n<d\le N$, are algebraically independent over $\bQ$ and have finite
orders $\rho_1,\ldots,\rho_d$.  Let $K$ be a number field of
degree~\hbox{$[K:\bQ]$}.  Suppose that the ring $K[f_1,\ldots,f_N]$ is
stable under all derivations $d/dz_1,\ldots,d/dz_n$. Then the set $S$
of points $z\in\bC^n$, distinct from the poles of the $F_j$'s, such
that $(F_1(z),\ldots,F_N(z))\in K^N$ is contained in an algebraic
hypersurface whose degree $\delta$ satisfies
$${\delta+n(n-1)/2\over n!}\le{\rho_1+\ldots+\rho_d\over d-n}[K:\bQ].$$
\endclaim

\proof{Proof} If the set $S$ is not contained in any algebraic hypersurface of
degree $<\delta$, the linear algebra argument used in the proof of
Cor.~10.6 shows that we can find $m=m(\delta)$ points $w_1,\ldots,w_m\in S$
which are not located on any algebraic hypersurface of degree~$<\delta$.
Let $H_1,\ldots,H_d$ be the denominators of $F_1,\ldots,F_d$. The standard
arithmetical methods of transcendental number theory allow us to
construct a sequence of entire functions in the following way: we set
$$G=P(F_1,\ldots,F_d)(H_1\ldots H_d)^s$$
where $P$ is a polynomial of degree $\le s$ in each variable with integer
coefficients. The polynomials $P$ are chosen so that $G$ vanishes at a very
high order at each point~$w_j$. This amounts to solving a linear system
whose unknowns are the coefficients of $P$ and whose coefficients
are polynomials in the derivatives of the $F_j$'s (hence lying
in the number field $K$). Careful estimates of size and denominators
and a use of the Dirichlet-Siegel box principle lead to the
following lemma, see e.g.\ \cite{Waldschmidt 1978}.
\endproof

\claim{(10.10) Lemma} For every $\varepsilon>0$, there exist constants
$C_1,C_2>0$, $r\ge 1$ and an infinite sequence $G_t$ of entire
functions, $t\in T\subset \bN$ $($depending on $m$ and on the choice
of the points $w_j)$, such that
\smallskip
\item{\rm a)} $G_t$ vanishes  at order $\ge t$ at all points $w_1,\ldots,w_m\,;$
\smallskip
\item{\rm b)} $|G_t|_r\ge (C_1t)^{-t\,[K:\bQ]}\,;$
\smallskip
\item{\rm c)} $|G_t|_{R(t)}\le C_2^t$~~ where
$R(t)=(t^{d-n}/\log t)^{1/(\rho_1+\ldots+\rho_d+\varepsilon)}$.
\smallskip
\endclaim

An application of Cor.~10.6 to $F=G_t$ and $R=R(t)$ gives the desired
bound for the degree $\delta$ as $t$ tends to~$+\infty$ and
$\varepsilon$ tends to~$0$. If $\delta_0$ is the largest integer which
satisfies the inequality of Th.~10.9, we get a contradiction if we take
$\delta=\delta_0+1$. This shows that $S$ must be contained in an
algebraic hypersurface of degree $\delta\le\delta_0$.\qed


\chapter{Chapter IV}{Sheaf Cohomology and Spectral Sequences}

\begineightpoint
One of the main topics of this book is the computation of various
cohomology groups arising in algebraic geometry.  The theory of
sheaves provides a general framework in which many cohomology theories
can be treated in a unified way.  The cohomology theory of sheaves will
be constructed here by means of Godement's simplicial flabby resolution. 
However, we have emphasized the analogy with Alexander-Spanier
cochains in order to give a simple definition of the cup product.  In
this way, all the basic properties of cohomology groups (long exact
sequences, Mayer Vietoris exact sequence, Leray's theorem, relations
with Cech cohomology, De Rham-Weil isomorphism theorem) can be derived
in a very elementary way from the definitions.  Spectral sequences and
hypercohomology groups are then introduced, with two principal examples
in view: the Leray spectral sequence and the Hodge-Fr\"olicher spectral
sequence.  The basic results concerning cohomology groups with constant
or locally constant coefficients (invariance by homotopy, Poincar\'e
duality, Leray-Hirsch theorem) are also included, in order to present a
self-contained approach of algebraic topology. 
\endeightpoint

\section{1}{Basic Results of Homological Algebra}
Let us first recall briefly some standard notations and results of homological
algebra that will be used systematically in the sequel.
Let $R$ be a commutative ring with unit.
A {\it differential module} $(K,d)$ is a $R$-module $K$ together with
an endomorphism $d:K\to K$, called the {\it differential}, such that 
$d\circ d=0$. The modules of {\it cycles} and of {\it boundaries} of $K$ are
defined respectively by
$$Z(K)=\ker d,~~~~B(K)=\Im d.\leqno(1.1)$$
Our hypothesis $d\circ d=0$ implies $B(K)\subset Z(K)$.
The {\it homology group} of $K$ is by definition the quotient module
$$H(K)=Z(K)/B(K).\leqno(1.2)$$
A {\it morphism of differential modules} $\varphi:K\longrightarrow L$ is a
$R$-homomorphism $\varphi:K\longrightarrow L$ such that $d\circ\varphi=
\varphi\circ d\,;$ here we denote by the same symbol $d$ the differentials
of $K$ and $L$. It is then clear that $\varphi\big(Z(K)\big)\subset Z(L)$
and $\varphi\big(B(K)\big)\subset B(L)$. Therefore, we get
an induced morphism on homology groups, denoted
$$H(\varphi):H(K)\longrightarrow H(L).\leqno(1.3)$$
It is easily seen that $H$ is a functor, i.e.\ $H(\psi\circ\varphi)=
H(\psi)\circ H(\varphi)$. We say that two morphisms $\varphi,\psi:K\longrightarrow L$
are {\it homotopic} if there exists a $R$-linear map $h:K\longrightarrow L$ such that
$$d\circ h+h\circ d=\psi-\varphi.\leqno(1.4)$$
Then $h$ is said to be a {\it homotopy} between $\varphi$
and $\psi$. For every cocycle $z\in Z(K)$, we infer
$\psi(z)-\varphi(z)=dh(z)$, hence the maps
$H(\varphi)$ and $H(\psi)$ coincide. The module $K$
itself is said to be homotopic to $0$ if $\Id_{K}$ is
homotopic to $0\,$; then $H(K)=0$.

\claim{(1.5) Snake lemma} Let 
$$0\longrightarrow K\buildo\varphi\over\longrightarrow L\buildo\psi
\over\longrightarrow M\longrightarrow 0$$
be a short exact sequence of morphisms of differential modules.
Then there exists a homomorphism $\partial:H(M)\longrightarrow H(K)$, called the
connecting homomorphism, and a homology exact sequence
$$\eqalign{
&H(K)\buildo H(\varphi)\over
\llra H(L)
\buildo H(\psi)\over
\llra H(M)\cr
&~~~~\raise2pt\hbox{$\nwarrow$}\kern-4pt
\buildo{\displaystyle\partial}\over{\hbox to 105pt{\hrulefill}}
\kern-3pt\raise2.5pt\hbox{\it /}\cr}$$
Moreover, to any commutative diagram of short exact sequences
$$\cmalign{
0\longrightarrow&~K~\longrightarrow&~L~\longrightarrow&~M~\longrightarrow 0\cr
\noalign{\vskip3pt}
&~\big\downarrow~&~\big\downarrow~&~\big\downarrow~\cr
\noalign{\vskip3pt}
0\longrightarrow&~\wt K~\longrightarrow&~\wt L~\longrightarrow&~\wt M~
\longrightarrow 0\cr}$$
is associated a commutative diagram of homology exact sequences
$$\cmalign{
&H(K)~\longrightarrow&~H(L)~\longrightarrow&~H(M)~\buildo
\partial\over\longrightarrow&~H(K)~\longrightarrow\cdots\cr
\noalign{\vskip3pt}
&~~~\big\downarrow&~~~\big\downarrow&~~~\big\downarrow&~~~\big\downarrow\cr
\noalign{\vskip3pt}
&H(\wt K)~\longrightarrow&~H(\wt L)~\longrightarrow&~H(\wt M)~\buildo
\partial\over\longrightarrow&~H(\wt K)~\longrightarrow\cdots.\cr}$$
\endclaim

\proof{Proof} We first define the connecting homomorphism $\partial$~:
let $m\in Z(M)$ represent a given cohomology class $\{m\}$
in $H(M)$. Then
$$\partial\{m\}=\{k\}\in H(K)$$
is the class of any element $k\in\varphi^{-1}d\psi^{-1}(m)$, as obtained
through the following construction:
$$\cmalign{
&&l\in L&\buildo{\displaystyle\psi}\over{\mapstex 24 }&~~m\in M\cr
\noalign{\vskip5pt}
&&~~
\buildo{\raise-0.2pt\hbox{$\begarrow$}}\over{\Big\downarrow}~d~~&&~~~
\buildo{\raise-0.2pt\hbox{$\begarrow$}}\over{\Big\downarrow}~d\cr
\noalign{\vskip-0.2pt}
\hfill k\in K~~&\buildo{\displaystyle\varphi}
\over{\mapstex 24 }~~&dl\in L&
\buildo{\displaystyle\psi}\over{\mapstex 24 }&~~0\in M.\cr}$$
The element $l$ is chosen to be a preimage of $m$ by the surjective map
$\psi$~; as $\psi(dl)=d(m)=0$, there exists 
a unique element $k\in K$ such that $\varphi(k)=dl$.
The element $k$ is actually a cocycle in $Z(K)$ because
$\varphi$ is injective and
$$\varphi(dk)=d\varphi(k)=
d(dl)=0~~\Longrightarrow~~dk=0.$$
The map $\partial$ will be well defined if we show that the cohomology
class $\{k\}$ depends only on $\{m\}$ and not on the choices made for
the representatives $m$ and~$l$. Consider another representative
$m'=m+dm_1$. Let $l_1\in L$ such that $\psi(l_1)=m_1$. Then $l$
has to be replaced by an element $l'\in L$ such that
$$\psi(l')=m+dm_1=\psi(l+dl_1).$$
It follows that $l'=l+dl_1+\varphi(k_1)$
for some $k_1\in K$, hence
$$dl'=dl+d\varphi(k_1)=\varphi(k)+\varphi(dk_1)=\varphi(k'),$$
therefore $k'=k+dk_1$ and $k'$ has the same cohomology 
class as $k$.

Now, let us show that $\ker\partial=\Im H(\psi)$.
If $\{m\}$ is in the image of $H(\psi)$, we can take
$m=\psi(l)$ with $dl=0$, thus $\partial\{m\}=0$.
Conversely, if $\partial\{m\}=\{k\}=0$, we have $k=dk_1$ for some
$k_1\in K$, hence $dl=\varphi(k)=d\varphi(k_1)$,
$z:=l-\varphi(k_1)\in Z(L)$ and $m=\psi(l)=\psi(z)$
is in $\Im H(\psi)$. We leave the verification of the
other equalities $\Im H(\varphi)=\ker H(\psi)$,
$\Im\partial=\ker H(\varphi)$ and of the commutation
statement to the reader.\qed
\endproof

In most applications, the differential modules come with a natural
$\bZ$-grading. A homological complex is a graded differential module
$K_\bu=\bigoplus_{q\in\bZ}K_q$ together with a differential $d$ of
degree $-1$, i.e.\ $d=\bigoplus d_q$ with $d_q:K_q\longrightarrow K_{q-1}$ and
$d_{q-1}\circ d_q=0$. Similarly, a cohomological complex is a
graded differential module $K^\bu=\bigoplus_{q\in\bZ}K^q$ with
differentials $d^q:K^q\longrightarrow K^{q+1}$ such that $d^{q+1}\circ d^q=0$
(superscripts are always used instead of subscripts in that case).
The corresponding (co)cycle, (co)boundary and (co)homology modules
inherit a natural $\bZ$-grading. In the case of cohomology, say, these
modules will be denoted
$$Z^\bu(K^\bu)=\bigoplus Z^q(K^\bu),~~
B^\bu(K^\bu)=\bigoplus B^q(K^\bu),~~H^\bu(K^\bu)=\bigoplus H^q(K^\bu).$$
Unless otherwise stated, morphisms of complexes are assumed to be of
degree $0$, i.e.\ of the form $\varphi^\bu=\bigoplus\varphi^q$ with
$\varphi^q:K^q\longrightarrow L^q$. Any short exact sequence
$$0\longrightarrow K^\bu\buildo\varphi^\bu\over\longrightarrow L^\bu\buildo\psi^\bu
\over\longrightarrow M^\bu\longrightarrow 0$$
gives rise to a corresponding {\it long exact sequence} of cohomology groups
$$H^q(K^\bu)\buildo H^q(\varphi^\bu)\over
\llra H^q(L^\bu)\buildo H^q(\psi^\bu)\over
\llra H^q(M^\bu)
\buildo\partial^q\over\longrightarrow H^{q+1}(K^\bu)\buildo H^{q+1}(\varphi^\bu)\over
{\relbar\joinrel\relbar\joinrel\longrightarrow\cdots}\leqno(1.6)$$
and there is a similar homology long exact sequence with a connecting
homomorphism $\partial_q$ of degree $-1$. When dealing with commutative
diagrams of such sequences, the following simple lemma is often useful;
the proof consists in a straightforward diagram chasing.

\claim{(1.7) Five lemma} Consider a commutative diagram of 
$R$-modules
$$\cmalign{
&~A_1~\hfill\longrightarrow&~A_2~\hfill\longrightarrow&~A_3~\hfill\longrightarrow&~A_4~\hfill\longrightarrow&~A_5\cr
\noalign{\vskip4pt}
&~\big\downarrow\varphi_1~&~\big\downarrow\varphi_2~&~\big\downarrow\varphi_3~
&~\big\downarrow\varphi_4~&~\big\downarrow\varphi_5~\cr
\noalign{\vskip4pt}
&~B_1~\hfill\longrightarrow&~B_2~\hfill\longrightarrow&~B_3~\hfill\longrightarrow&~B_4~\hfill\longrightarrow&~B_5\cr}$$
where the rows are exact sequences. If $\varphi_2$ and $\varphi_4$
are injective and $\varphi_1$ surjective, then $\varphi_3$ is
injective. If $\varphi_2$ and $\varphi_4$ is surjective and $\varphi_5$
injective, then $\varphi_3$ is surjective. In particular, $\varphi_3$
is an isomorphism as soon as $\varphi_1,\varphi_2,\varphi_4,\varphi_5$
are isomorphisms.
\endclaim

\section{2}{The Simplicial Flabby Resolution of a Sheaf}
Let $X$ be a topological space and let $\cA$ be a sheaf of abelian groups
on~$X$ (see \S~II-2 for the definition). All the sheaves appearing in
the sequel are assumed implicitly to be sheaves of {\it abelian groups},
unless otherwise stated. The first useful notion is that of resolution.

\claim{(2.1) Definition} A $($cohomological$)$ resolution of $\cA$ is a
differential complex of sheaves $(\cL^\bu,d)$ with
$\cL^q=0$, $d^q=0$ for $q<0$, such that there is an exact sequence
$$0\longrightarrow\cA\buildo j\over\longrightarrow\cL^0\buildo d^0\over\longrightarrow\cL^1\longrightarrow\cdots
\longrightarrow\cL^q\buildo d^q\over\longrightarrow\cL^{q+1}\longrightarrow\cdots~.$$
If $\varphi:\cA\longrightarrow\cB$ is a morphism of sheaves
and $(\cM^\bu,d)$ a resolution of~$\cB$, a morphism of resolutions
$\varphi^\bu:\cL^\bu\longrightarrow\cM^\bu$ is a commutative diagram
$$\cmalign{
0\longrightarrow&~\cA~\buildo j\over\longrightarrow&~\cL^0~\hfill\buildo d^0\over\longrightarrow&~\cL^1~\hfill\longrightarrow\cdots
\longrightarrow&~\cL^q~\hfill\buildo d^q\over\longrightarrow&~\cL^{q+1}~
\hfill\longrightarrow&~\cdots\cr
\noalign{\vskip4pt}
&~\big\downarrow\varphi~&~\big\downarrow\varphi^0~
&~\big\downarrow\varphi^1~&~\big\downarrow\varphi^q~
&~\big\downarrow\varphi^{q+1}~&\cr
0\longrightarrow&~\cB~\hfill\buildo j\over\longrightarrow&~\cM^0~
\buildo d^0\over\longrightarrow&~\cM^1~\longrightarrow\cdots
\longrightarrow&~\cM^q~\buildo d^q\over\longrightarrow&~\cM^{q+1}~
\hfill\longrightarrow&~\cdots~.\cr}$$
\endclaim

\claim{(2.2) Example} \rm Let $X$ be a differentiable manifold and $\cE^q$ 
the sheaf of germs of $\cC^\infty$ differential forms of degree $q$
with real values. The exterior derivative $d$ defines a
resolution $(\cE^\bu,d)$ of the sheaf $\bR$ of locally constant functions
with real values. In fact Poincar\'e's lemma asserts that $d$ is
locally exact in degree $q\ge 1$, and it is clear that the sections of
$\ker d^0$ on connected open sets are constants.\qed
\endclaim

In the sequel, we will be interested by special resolutions in which the 
sheaves $\cL^q$ have no local ``rigidity''. For that purpose, we introduce
flabby sheaves, which have become a standard tool in sheaf theory
since the publication of Godement's book \cite{Godement 1957}.

\claim{(2.3) Definition} A sheaf $\cF$ is called flabby if for every open 
subset $U$ of $X$, the restriction map $\cF(X)\longrightarrow\cF(U)$
is onto, i.e.\ if every section of $\cF$ on $U$ can be extended to $X$.
\endclaim

Let $\pi:\cA\longrightarrow X$ be a sheaf on $X$.  We denote by
$\cA^{[0]}$ the sheaf of germs of sections $X\longrightarrow\cA$ which
are {\it not necessarily continuous}.  In other words, $\cA^{[0]}(U)$
is the set of all maps $f:U\longrightarrow\cA$ such that
$f(x)\in\cA_x$ for all $x\in U$, or equivalently
$\cA^{[0]}(U)=\prod_{x\in U}\cA_x$.  It is clear that $\cA^{[0]}$ is
flabby and there is a canonical injection
$$j:\cA\longrightarrow\cA^{[0]}$$
defined as follows: to any $s\in\cA_x$ we associate the germ
$\wt s\in\cA^{[0]}_x$ equal to the
continuous section $y\longmapsto\wt s(y)$ near $x$ such that 
$\wt s(x)=s$. In the sequel we merely denote 
$\wt s:y\longmapsto s(y)$ for simplicity. The sheaf
$\cA^{[0]}$ is called the {\it canonical flabby sheaf} associated to
$\cA$. We define inductively 
$$\cA^{[q]}=(\cA^{[q-1]})^{[0]}.$$ 
The stalk $\cA^{[q]}_x$ can be considered as the set of equivalence classes of
maps $f:X^{q+1}\longrightarrow\cA$ such that $f(x_0,\ldots,x_q)\in\cA_{x_q}$, with two such
maps identified if they coincide on a set of the form
$$x_0\in V,~~~x_1\in V(x_0),~~\ldots~,~~~x_q\in V(x_0,\ldots,x_{q-1}),
\leqno(2.4)$$
where $V$ is an open neighborhood of $x$ and $V(x_0,\ldots,x_j)$ an open
neighborhood of $x_j$, depending on $x_0,\ldots,x_j$. This is 
easily seen by induction on~$q$, if we identify a map $f:X^{q+1}\to\cA$
to the map $X\to\cA^{[q-1]}$, $x_0\mapsto f_{x_0}$ such that
$f_{x_0}(x_1,\ldots,x_q)=f(x_0,x_1,\ldots,x_q)$. Similarly,
$\cA^{[q]}(U)$ is the set of equivalence classes of functions
$X^{q+1}\ni(x_0,\ldots,x_q)\longmapsto f(x_0,\ldots,x_q)\in\cA_{x_q}$,
with two such functions identified if they coincide on a set of the form
$$x_0\in U,~~~x_1\in V(x_0),~~\ldots~,~~~x_q\in V(x_0,\ldots,x_{q-1}).
\leqno(2.4')$$
Here, we may of course suppose $V(x_0,\ldots,x_{q-1})\subset\ldots\subset
V(x_0,x_1)\subset V(x_0)\subset U$. We define a differential 
$d^q:\cA^{[q]}\longrightarrow\cA^{[q+1]}$ by
$$\leqalignno{
&(d^qf)(x_0,\ldots,x_{q+1})=&(2.5)\cr
&\sum_{0\le j\le q}(-1)^jf(x_0,\ldots,\wh{x_j},\ldots,x_{q+1})
+(-1)^{q+1}f(x_0,\ldots,x_q)(x_{q+1}).\cr}$$
The meaning of the last term is to be understood as follows:
the element $s=f(x_0,\ldots,x_q)$ is a germ in $\cA_{x_q}$, therefore $s$
defines a continuous section $x_{q+1}\mapsto s(x_{q+1})$ of $\cA$ in a
neighborhood $V(x_0,\ldots,x_q)$ of $x_q$. In low degrees, we have the 
formulas
$$\leqalignno{
(js)(x_0)&=s(x_0),~~~s\in\cA_x,\cr
(d^0f)(x_0,x_1)&=f(x_1)-f(x_0)(x_1),~~~f\in\cA^{[0]}_x,&(2.6)\cr
(d^1f)(x_0,x_1,x_2)&=f(x_1,x_2)-f(x_0,x_2)+f(x_0,x_1)(x_2),
~~~f\in\cA^{[1]}_x.\cr}$$

\claim{(2.7) Theorem {\rm (\cite{Godement 1957})}} The complex
$(\cA^{[\bu]},d)$ is a resolution of the sheaf $\cA$, called the 
simplicial flabby resolution of $\cA$.
\endclaim

\proof{Proof} For $s\in\cA_x$, the associated continuous germ obviously
satisfies $s(x_0)(x_1)=s(x_1)$
for $x_0\in V$, $x_1\in V(x_0)$ small enough.
The reader will easily infer from this that $d^0\circ j=0$ and
$d^{q+1}\circ d^q=0$. In order to verify that
$(\cA^{[\bu]},d)$ is a resolution of $\cA$, we show that the complex
$$\cdots\longrightarrow 0\longrightarrow\cA_x\buildo j\over\longrightarrow\cA^{[0]}_x\buildo d^0\over\longrightarrow
\cdots\longrightarrow\cA^{[q]}_x\buildo d^q\over\longrightarrow\cA^{[q+1]}_x\longrightarrow\cdots$$
is homotopic to zero for every point $x\in X$. Set $\cA^{[-1]}=\cA$,
$d^{-1}=j$ and
$$\cmalign{
&h^0~:~~\cA^{[0]}_x\longrightarrow\cA_x,~~~~&h^0(f)=f(x)\in\cA_x,\cr
&h^q~:~~\cA^{[q]}_x\longrightarrow\cA^{[q-1]}_x,~~~~&h^q(f)(x_0,\ldots,x_{q-1})=
f(x,x_0,\ldots,x_{q-1}).\cr}$$
A straightforward computation shows that
$(h^{q+1}\circ d^q+d^{q-1}\circ h^q)(f)=f$ for all $q\in\bZ$ and $f\in
\cA^{[q]}_x$.\qed
\endproof

If $\varphi:\cA\longrightarrow\cB$ is a sheaf morphism, it is clear that $\varphi$
induces a morphism of resolutions
$$\varphi^{[\bu]}:\cA^{[\bu]}\longrightarrow\cB^{[\bu]}.\leqno(2.8)$$
For every short exact sequence $\cA\to\cB\to\cC$ of sheaves,
we get a corresponding short exact sequence of sheaf complexes
$$\cA^{[\bu]}\longrightarrow\cB^{[\bu]}\longrightarrow\cC^{[\bu]}.\leqno(2.9)$$

\section{3}{Cohomology Groups with Values in a Sheaf}
\subsection{3.A}{Definition and Functorial Properties}
If $\pi:\cA\to X$ is a sheaf of abelian groups, the {\it cohomology groups} of
$\cA$ on $X$ are (in a vague sense) algebraic invariants 
which describe the rigidity properties of the global sections of $\cA$. 

\claim{(3.1) Definition} For every $q\in\bZ$, the $q$-th cohomology 
group of $X$ with values in $\cA$ is
$$\eqalign{
H^q(X&,\cA)=H^q\big(\cA^{[\bu]}(X)\big)=\cr
&=\ker\big(d^q:\cA^{[q]}(X)\to\cA^{[q+1]}(X)\big)/
  \Im(d^{q-1}:\cA^{[q-1]}(X)\to\cA^{[q]}(X)\big)\cr}$$
with the convention $\cA^{[q]}=0$, $d^q=0$, $H^q(X,\cA)=0$ when $q<0$.
\endclaim

For any subset $S\subset X$, we denote by $\cA_{\restriction S}$
the {\it restriction} of $\cA$ to $S$, i.e.\ the sheaf
$\cA_{\restriction S}=\pi^{-1}(S)$ equipped with the projection
$\pi_{\restriction S}$ onto $S$. Then we write $H^q(S,
\cA_{\restriction S})=H^q(S,\cA)$ for simplicity.
When $U$ is open, we see that $(\cA^{[q]})_{\restriction U}$ coincides with
$(\cA_{\restriction U})^{[q]}$, thus we have $H^q(U,\cA)=
H^q\big(\cA^{[\bu]}(U)\big)$.
It is easy to show that every exact sequence of sheaves 
$0\to\cA\to\cL^0\to\cL^1$ induces an exact sequence
$$0\longrightarrow\cA(X)\longrightarrow\cL^0(X)\longrightarrow\cL^1(X).\leqno(3.2)$$
If we apply this to $\cL^q=\cA^{[q]}$, $q=0,1$, we conclude that
$$H^0(X,\cA)=\cA(X).\leqno(3.3)$$

Let $\varphi:\cA\longrightarrow\cB$ be a sheaf morphism; (2.8) shows that there is
an induced morphism
$$H^q(\varphi):H^{q}(X,\cA)\longrightarrow H^q(X,\cB)\leqno(3.4)$$
on cohomology groups. Let $0\to\cA\to\cB\to\cC\to 0$ be an exact sequence 
of sheaves. Then we have an exact sequence of {\it groups}
$$0\longrightarrow\cA^{[0]}(X)\longrightarrow\cB^{[0]}(X)\longrightarrow\cC^{[0]}(X)\longrightarrow0$$
because $\cA^{[0]}(X)=\prod_{x\in X}\cA_x$. Similarly, (2.9) yields
for every $q$ an exact sequence of {\it groups}
$$0\longrightarrow\cA^{[q]}(X)\longrightarrow\cB^{[q]}(X)\longrightarrow\cC^{[q]}(X)\longrightarrow0.$$
If we take (3.3) into account, the snake lemma implies:

\claim{(3.5) Theorem} To any exact sequence of sheaves
$0\to\cA\to\cB\to\cC\to 0$ is associated a long exact sequence of
cohomology groups
$$\cmalign{
\noalign{\vskip3pt}
0~&~\longrightarrow~~\cA(X)&~\longrightarrow~~\cB(X)&~\longrightarrow~~\cC(X)&~\longrightarrow~~H^1(X,\cA)&~\longrightarrow~\cdots\cr
\noalign{\vskip3pt}
\cdots&~\longrightarrow~H^q(X,\cA)&~\longrightarrow~H^q(X,\cB)&~\longrightarrow~H^q(X,\cC)&~\longrightarrow~H^{q+1}(X,\cA)
&~\longrightarrow~\cdots.\cr}$$
\endclaim

\claim{(3.6) Corollary} Let $\cB\to\cC$ be a surjective sheaf morphism and
let $\cA$ be its kernel. If $H^1(X,\cA)=0$, then $\cB(X)\to\cC(X)$
is surjective.\qed
\endclaim

\subsection{3.B}{Exact Sequence Associated to a Closed Subset}
Let $S$ be a closed subset of $X$ and $U=X\ssm S$. For any 
sheaf $\cA$ on $X$, the presheaf
$$\Omega\longmapsto\cA(S\cap\Omega),~~~\Omega\subset X~~\hbox{\rm open}$$
with the obvious restriction maps satisfies axioms (II-$2.4')$ and
(II-$2.4'')$, so it defines a sheaf on $X$ which we denote by $\cA^S$.
This sheaf should not be confused with the restriction sheaf 
$\cA_{\restriction S}$, which is a sheaf on $S$. We easily find
$$(\cA^S)_x=\cA_x~~~\hbox{\rm if}~~x\in S,~~~(\cA^S)_x=0~~~\hbox{\rm if}~~x\in U.
\leqno(3.7)$$
Observe that these relations would completely fail if $S$ were not closed.
The restriction morphism $f\mapsto f_{\restriction S}$ induces a surjective
sheaf morphism $\cA\to\cA^S$. We let $\cA_U$ be its kernel,
so that we have the relations
$$(\cA_U)_x=0~~~\hbox{\rm if}~~x\in S,~~~(\cA_U)_x=\cA_x~~~\hbox{\rm if}~~x\in U.
\leqno(3.8)$$
From the definition, we obtain in particular
$$\cA^S(X)=\cA(S),~~~\cA_U(X)=\{\hbox{\rm sections of}~\cA(X)~
\hbox{\rm vanishing on}~S\}.\leqno(3.9)$$
Theorem 3.5 applied to the exact sequence $0\to\cA_U\to\cA\to\cA^S\to 0$ 
on $X$ gives a long exact sequence
$$\cmalign{
\noalign{\vskip3pt}
0&~\longrightarrow~~~\cA_U(X)&~\longrightarrow~~~\cA(X)&~\longrightarrow~~~\cA(S)&~\longrightarrow~~~H^1(X,\cA_U)&~\cdots\cr
\noalign{\vskip3pt}
&~\longrightarrow~ H^q(X,\cA_U)&~\longrightarrow~ H^q(X,\cA)&~\longrightarrow~ H^q(X,\cA^S)&~\longrightarrow~ H^{q+1}(X,\cA_U)
&~\cdots\cr}\leqno{\displaystyle(3.9)\atop\displaystyle\strut}$$

\subsection{3.C}{Mayer-Vietoris Exact Sequence}
Let $U_1$, $U_2$ be open subsets of $X$ and $U=U_1\cup U_2$, $V=U_1\cap U_2$.
For any sheaf $\cA$ on $X$ and any $q$ we have an exact sequence
$$0\longrightarrow\cA^{[q]}(U)\longrightarrow\cA^{[q]}(U_1)\oplus\cA^{[q]}(U_2)\longrightarrow
\cA^{[q]}(V)\longrightarrow 0$$
where the injection is given by $f\longmapsto(f_{\restriction U_1},
f_{\restriction U_2})$ and the surjection by
$(g_1,g_2)\longmapsto g_{2\restriction V}-g_{1\restriction V}$~; 
the surjectivity of this map follows immediately from the fact that $\cA^{[q]}$
is flabby. An application of the snake lemma yields:

\claim{(3.11) Theorem} For any sheaf $\cA$ on $X$ and any open sets 
$U_1,U_2\subset X$, set $U=U_1\cup U_2$, $V=U_1\cap U_2$. Then
there is an exact sequence
$$H^q(U,\cA)\longrightarrow H^q(U_1,\cA)\oplus H^q(U_2,\cA)\longrightarrow H^q(V,\cA)
\longrightarrow H^{q+1}(U,\cA)\cdots\eqno\square$$
\endclaim

\section{4}{Acyclic Sheaves}
Given a sheaf $\cA$ on $X$, it is usually very important to decide
whether the cohomology groups $H^q(U,\cA)$ vanish for $q\ge 1$, and if
this is the case, for which type of open sets~$U$. Note that one cannot
expect to have $H^0(U,\cA)=0$ in general, since a sheaf always has
local sections.

\claim{(4.1) Definition} A sheaf $\cA$ is said to be acyclic on an open
subset $U$ if $H^q(U,\cA)=0$ for $q\ge 1$.
\endclaim

\subsection{4.A}{Case of Flabby Sheaves}
We are going to show that flabby sheaves are acyclic. First we need the
following simple result.

\claim{(4.2) Proposition} Let $\cA$ be a sheaf with the following
property: for every section $f$ of $\cA$ on an open subset $U\subset X$
and every point $x\in X$, there exists a neighborhood $\Omega$ of $x$
and a section $h\in\cA(\Omega)$ such that $h=f$ on $U\cap\Omega$.  Then
$\cA$ is flabby.
\endclaim

A consequence of this proposition is that flabbiness is a local property: 
a sheaf $\cA$ is flabby on $X$ if and only if it is flabby
on a neighborhood of every point of $X$.

\proof{Proof} Let $f\in\cA(U)$ be given. Consider the set of pairs $(v,V)$
where $v$ in $\cB(V)$ is an extension of $f$ on an open subset $V\supset U$.
This set is inductively ordered, so there exists a maximal extension
$(v,V)$ by Zorn's lemma. The assumption shows that $V$ must be
equal to $X$.\qed
\endproof

\claim{(4.3) Proposition} Let $0\longrightarrow\cA\buildo j\over\longrightarrow\cB\buildo p
\over\longrightarrow\cC\longrightarrow 0$ be an exact sequence of sheaves. If $\cA$ is flabby, the
sequence of groups
$$0\longrightarrow\cA(U)\buildo j\over\longrightarrow\cB(U)\buildo p\over\longrightarrow\cC(U)\longrightarrow 0$$
is exact for every open set $U$. If $\cA$ and $\cB$ are flabby, then
$\cC$ is flabby.
\endclaim

\proof{Proof} Let $g\in\cC(U)$ be given. Consider the set $E$ of pairs $(v,V)$
where $V$ is an open subset of $U$ and $v\in\cB(V)$ is such that $p(v)=g$
on $V$. It is clear that $E$ is inductively ordered, so
$E$ has a maximal element $(v,V)$, and we will prove
that $V=U$. Otherwise, let $x\in U\ssm V$ and let $h$ be a section
of $\cB$ in a neighborhood of $x$ such that $p(h_x)=g_x$. Then
$p(h)=g$ on a neighborhood $\Omega$ of $x$, thus $p(v-h)=0$ on 
$V\cap\Omega$ and $v-h=j(u)$ with $u\in\cA(V\cap\Omega)$. If $\cA$ is
flabby, $u$ has an extension $\wt u\in\cA(X)$ and we can define 
a section $w\in\cB(V\cup\Omega)$ such that $p(w)=g$ by
$$w=v~~\hbox{\rm on}~~V,~~~~w=h+j(\wt u)~~\hbox{\rm on}~~\Omega,$$
contradicting the maximality of $(v,V)$.
Therefore $V=U$, $v\in\cB(U)$ and $p(v)=g$ on $U$. The first statement is
proved. If $\cB$ is also flabby, $v$ has an extension $\wt v\in\cB(X)$ and
$\wt g=p(\wt v)\in\cC(X)$ is an extension of $g$. Hence $\cC$ is 
flabby.\qed
\endproof

\claim{(4.4) Theorem} A flabby sheaf $\cA$ is acyclic on all open sets
$U\subset X$.
\endclaim

\proof{Proof} Let $\cZ^q=\ker\big(d^q:\cA^{[q]}\to\cA^{[q+1]}\big)$. Then 
$\cZ^0=\cA$ and we have an exact sequence of sheaves
$$0\longrightarrow\cZ^q\longrightarrow\cA^{[q]}\buildo d^q\over\longrightarrow\cZ^{q+1}\longrightarrow 0$$
because $\Im d^q=\ker d^{q+1}=\cZ^{q+1}$. Proposition 4.3 implies by
induction on $q$ that all sheaves $\cZ^q$ are flabby, and yields exact
sequences
$$0\longrightarrow\cZ^q(U)\longrightarrow\cA^{[q]}(U)\buildo d^q\over\longrightarrow\cZ^{q+1}(U)\longrightarrow 0.$$
For $q\ge 1$, we find therefore
$$\eqalign{
\ker\big(d^q:\cA^{[q]}(U)\to\cA^{[q+1]}(U)\big)&=\cZ^q(U)\cr
&=\Im\big(d^{q-1}:\cA^{[q-1]}(U)\to\cA^{[q]}(U)\big),\cr}$$
that is, $H^q(U,\cA)=H^q\big(\cA^{[\bu]}(U)\big)=0$.\qed
\endproof

\subsection{4.B}{Soft Sheaves over Paracompact Spaces}
We now discuss another general situation which produces acyclic sheaves.
Recall that a topological space $X$ is said to be {\it paracompact} 
if $X$ is Hausdorff and if every open covering of $X$ has a
locally finite refinement. For instance, it is well known that every metric
space is paracompact. A paracompact space $X$ is always {\it normal}\/;
in particular, for any locally finite open covering $(U_\alpha)$ of $X$ there 
exists an open covering $(V_\alpha)$ such that $\ol V_\alpha\subset 
U_\alpha$. We will also need another closely related concept.

\claim{(4.5) Definition} We say that a subspace $S$ is strongly 
paracompact in $X$ if $S$ is Hausdorff and if the following property is
satisfied: for every covering $(U_\alpha)$ of $S$ by open sets in $X$, 
there exists another such covering $(V_\beta)$ and a neighborhood $W$ of $S$
such that each set $W\cap\ol V_\beta$ is contained in some $U_\alpha$, 
and such that every point of $S$ has a neighborhood intersecting only 
finitely many sets $V_\beta$.
\endclaim

It is clear that a strongly paracompact subspace $S$ is itself paracompact.
Conversely, the following result is easy to check:

\claim{(4.6) Lemma} A subspace $S$ is strongly paracompact in $X$ as soon as 
one of the following situations occurs:
\medskip
\item{\rm a)} $X$ is paracompact and $S$ is closed;
\smallskip
\item{\rm b)} $S$ has a fundamental family of paracompact neighborhoods 
in $X\,$;
\smallskip
\item{\rm c)} $S$ is paracompact and has a neighborhood homeomorphic
to some product $S\times T$, in which $S$ is embedded as a slice 
$S\times\{t_0\}$.\qed
\endclaim

\claim{(4.7) Theorem} Let $\cA$ be a sheaf on $X$ and $S$ a strongly
paracompact subspace of $X$. Then every section $f$ of $\cA$ on $S$ 
can be extended to a section of $\cA$ on some open neighborhood 
$\Omega$ of $\cA$.
\endclaim

\proof{Proof} Let $f\in\cA(S)$. For every point $z\in S$ there exists
an open neighborhood $U_z$ and a section $\smash{\wt f}_z\in\cA(U_z)$ such that 
$\smash{\wt f}_z(z)=f(z)$. After shrinking $U_z$,
we may assume that $\smash{\wt f}_z$ and $f$ coincide on $S\cap U_z$.
Let $(V_\alpha)$ be an open covering of $S$ that is locally finite near $S$ 
and $W$ a neighborhood of $S$ such that 
$W\cap\ol V_\alpha\subset U_{z(\alpha)}$ (Def.\ 4.5). We let
$$\Omega=\big\{x\in W\cap\bigcup V_\alpha\,;\,
\smash{\wt f}_{z(\alpha)}(x)=\smash{\wt f}_{z(\beta)}(x),~\forall
\alpha,\beta~\hbox{\rm with~}x\in\ol V_\alpha\cap\ol V_\beta\big\}.$$
Then $(\Omega\cap V_\alpha)$ is an open covering of $\Omega$ and all
pairs of sections $\smash{\wt f}_{z(\alpha)}$ coincide in pairwise
intersections.  Thus there exists a section $F$ of $\cA$ on $\Omega$
which is equal to $\smash{\wt f}_{z(\alpha)}$ on $\Omega\cap V_\alpha$. 
It remains only to show that $\Omega$ is a neighborhood of $S$.  Let
$z_0\in S$.  There exists a neighborhood $U'$ of $z_0$ which meets only
finitely many sets $V_{\alpha_1},\ldots,V_{\alpha_p}$.  After shrinking
$U'$, we may keep only those $V_{\alpha_l}$ such that $z_0\in\ol
V_{\alpha_l}$.  The sections $\smash{\wt f}_{z(\alpha_l)}$ coincide at
$z_0$, so they coincide on some neighborhood $U''$ of this point. 
Hence $W\cap U''\subset\Omega$, so $\Omega$ is a neighborhood of
$S$.\qed
\endproof

\claim{(4.8) Corollary} If $X$ is paracompact, every section $f\in\cA(S)$ 
defined on a closed set $S$ extends to a neighborhood $\Omega$ of
$S$.\qed
\endclaim

\claim{(4.9) Definition} A sheaf $\cA$ on $X$ is said to be soft if 
every section $f$ of $\cA$ on a closed set $S$ can be extended to $X$,
i.e.\ if the restriction map $\cA(X)\longrightarrow\cA(S)$ is onto for
every closed set $S$.
\endclaim

\claim{(4.10) Example} \rm On a paracompact space, every flabby sheaf 
$\cA$ is soft: this is a consequence of Cor.\ 4.8.
\endclaim

\claim{(4.11) Example} \rm On a paracompact space, the Tietze-Urysohn extension 
theorem shows that the sheaf $\cC_X$ of germs of continuous functions on $X$
is a soft sheaf of rings. However, observe that $\cC_X$ is not flabby 
as soon as $X$ is not discrete.
\endclaim

\claim{(4.12) Example} \rm If $X$ is a paracompact differentiable manifold, the 
sheaf $\cE_X$ of germs of $\cC^\infty$ functions on $X$ is a soft sheaf of 
rings.\qed
\endclaim

Until the end of this section, we assume that $X$ is a {\it paracompact 
topo\-lo\-gical space}. We first show that softness is a local property.

\claim{(4.13) Proposition} A sheaf $\cA$ is soft on $X$ if and only if
it is soft in a neighborhood of every point $x\in X$.
\endclaim

\proof{Proof} If $\cA$ is soft on $X$, it is soft on any closed neighborhood
of a given point. Conversely, let $(U_\alpha)_{\alpha\in I}$ be a locally 
finite open covering of $X$ which refine some covering by neighborhoods 
on which $\cA$ is soft. Let $(V_\alpha)$ be a finer covering such that 
$\ol V_\alpha\subset U_\alpha$, and $f\in\cA(S)$ be a section of $\cA$ 
on a closed subset $S$ of $X$. We consider the set $E$ of pairs $(g,J)$, 
where $J\subset I$ and where $g$ is a section over 
$F_J:=S\cup\bigcup_{\alpha\in J}\ol V_\alpha$, such that $g=f$ on $S$.
As the family $(\ol V_\alpha)$ is locally finite, a section of $\cA$
over $F_J$ is continuous as soon it is continuous on $S$ and on each 
$\ol V_\alpha$. Then $(f,\emptyset)\in E$ and $E$ is 
inductively ordered by the relation
$$(g',J')\longrightarrow(g'',J'')~~~\hbox{\rm if}~~J'\subset J''~~\hbox{\rm and}~~
g'=g''~~\hbox{\rm on}~~F_{J'}$$
No element $(g,J)$, $J\ne I$, can be maximal: the assumption shows
that $\smash{g_{\restriction F_J\cap\ol V_\alpha}}$ has an extension to 
$\ol V_\alpha$, thus such a $g$ has an extension to $F_{J\cup\{\alpha\}}$
for any $\alpha\notin J$. Hence $E$ has a maximal element $(g,I)$ 
defined on $F_I=X$.\qed
\endproof

\claim{(4.14) Proposition} Let $0\to\cA\to\cB\to\cC\to 0$ be an exact 
sequence of sheaves. If $\cA$ is soft, the map $\cB(S)\to\cC(S)$ is onto 
for any closed subset $S$ of $X$. If $\cA$ and $\cB$ are soft, 
then $\cC$ is soft.
\endclaim

By the above inductive method, this result can be proved in a way
similar to its analogue for flabby sheaves. We therefore obtain:

\claim{(4.15) Theorem} On a paracompact space, a soft sheaf is acyclic
on all closed subsets.\qed
\endclaim

\claim{(4.16) Definition} The {\it support} of a section 
$f\in\cA(X)$ is defined by 
$$\Supp\,f=\big\{x\in X\,;\,f(x)\ne 0\big\}.$$
\endclaim

$\Supp\,f$ is always a closed set: as $\cA\to X$ is a local 
homeomorphism, the equality $f(x)=0$ implies $f=0$ in a neighborhood of $x$.

\claim{(4.17) Theorem} Let $(U_\alpha)_{\alpha\in I}$ be an open covering of
$X$.  If $\cA$ is soft and $f\in\cA(X)$, there exists a partition of $f$
subordinate to $(U_\alpha)$, i.e.\ a family of sections
$f_\alpha\in\cA(X)$ such that $(\Supp\,f_\alpha)$ is locally finite,
$\Supp\,f_\alpha\subset U_\alpha$ and $\sum f_\alpha=f$ on $X$.
\endclaim

\proof{Proof} Assume first that $(U_\alpha)$ is locally finite.
There exists an open covering $(V_\alpha)$ such that
$\ol V_\alpha\subset U_\alpha$. Let $(f_\alpha)_{\alpha\in J}$, 
$J\subset I$, be a maximal family of sections $f_\alpha\in\cA(X)$ such 
that $\Supp\,f_\alpha\subset U_\alpha$ and $\sum_{\alpha\in J}f_\alpha=f$
on $S=\bigcup_{\alpha\in J}\ol V_\alpha$. If $J\ne I$ and $\beta\in
I\ssm J$, there exists a section $f_\beta\in\cA(X)$ such that
$$f_\beta=0~~~\hbox{\rm on}~~X\ssm U_\beta~~~\hbox{\rm and}~~~
f_\beta=f-\sum_{\alpha\in J}f_\alpha~~~\hbox{\rm on}~~S\cup\ol V_\beta$$
because $(X\ssm U_\beta)\cup S\cup\ol V_\beta$ is closed and
$f-\sum f_\alpha=0$ on $(X\ssm U_\alpha)\cap S$. This is a contradiction
unless $J=I$.

In general, let $(V_j)$ be a locally finite refinement
of $(U_\alpha)$, such that $V_j\subset U_{\rho(j)}$, and let
$(f'_j)$ be a partition of $f$ subordinate to $(V_j)$. Then
$f_\alpha=\sum_{j\in\rho^{-1}(\alpha)}f'_j$ is
the required partition of $f$.\qed
\endproof

Finally, we discuss a special situation which occurs very often in 
practice. Let $\cR$ be a sheaf of commutative rings on $X$~; the rings
$\cR_x$ are supposed to have a unit element. Assume that $\cA$ is
a sheaf of modules over $\cR$. It is clear that $\cA^{[0]}$
is a $\cR^{[0]}$-module, and thus also a $\cR$-module. Therefore
all sheaves $\cA^{[q]}$ are $\cR$-modules and the cohomology groups
$H^q(U,\cA)$ have a natural structure of $\cR(U)$-module.

\claim{(4.18) Lemma} If $\cR$ is soft, every sheaf $\cA$ of $\cR$-modules
is soft.
\endclaim

\proof{Proof} Every section $f\in\cA(S)$ defined on a closed set $S$
has an extension to some open neighborhood $\Omega$. Let $\psi\in\cR(X)$
be such that $\psi=1$ on $S$ and $\psi=0$ on $X\ssm\Omega$.
Then $\psi f$, defined as $0$ on $X\ssm\Omega$, is an extension of 
$f$ to~$X$.\qed
\endproof

\claim{(4.19) Corollary} Let $\cA$ be a sheaf of $\cE_X$-modules
on a paracompact differentiable manifold $X$. Then
$H^q(X,\cA)=0$ for all $q\ge 1$.
\endclaim

\section{5}{\v Cech Cohomology}
\subsection{5.A}{Definitions}
In many important circumstances, cohomology groups with values 
in a sheaf $\cA$ can be computed by means of the complex of \v Cech cochains,
which is directly related to the spaces of sections of $\cA$ on sufficiently 
fine coverings of $X$. This more concrete approach was historically the first
one used to define sheaf cohomology (\cite{Leray 1950}, \cite{Cartan 1950}); 
however \v Cech cohomology does not always coincide with the ``good" 
cohomology on non paracompact spaces. Let $\cU=(U_\alpha)_{\alpha\in I}$ be
an open covering of $X$. For the sake of simplicity, we denote
$$U_{\alpha_0\alpha_1\ldots\alpha_q}=U_{\alpha_0}\cap U_{\alpha_1}\cap
\ldots\cap U_{\alpha_q}.$$
The group $C^q(\cU,\cA)$ of {\it \v Cech $q$-cochains} is the set
of families 
$$c=(c_{\alpha_0\alpha_1\ldots\alpha_q})\in\prod_{(\alpha_0,\ldots,\alpha_q)
\in I^{q+1}}\cA(U_{\alpha_0\alpha_1\ldots\alpha_q}).$$
The group structure on $C^q(\cU,\cA)$ is the obvious one deduced from the
addition law on sections of $\cA$. The {\it \v Cech differential}
$\delta^q:C^q(\cU,\cA)\longrightarrow C^{q+1}(\cU,\cA)$ is defined by 
the formula
$$(\delta^qc)_{\alpha_0\ldots\alpha_{q+1}}=
\sum_{0\le j\le q+1}(-1)^j\,c_{\alpha_0\ldots\wh{\alpha_j}\ldots\alpha_{q+1}\;
\restriction U_{\alpha_0\ldots\alpha_{q+1}}},\leqno(5.1)$$
and we set $C^q(\cU,\cA)=0$, $\delta^q=0$ for $q<0$. In degrees $0$ and 
$1$, we get for example
$$\leqalignno{
&q=0,~~~c=(c_\alpha),~~~\hfill(\delta^0c)_{\alpha\beta}
=c_\beta-c_{\alpha\;\restriction U_{\alpha\beta}},&(5.2)\cr
&q=1,~~~c=(c_{\alpha\beta}),~~~(\delta^1c)_{\alpha\beta\gamma}
=c_{\beta\gamma}-c_{\alpha\gamma}+c_{\alpha\beta\;
\restriction U_{\alpha\beta\gamma}}.&(5.2')\cr}$$
Easy verifications left to the reader show that $\delta^{q+1}\circ\delta^q=0$.
We get therefore a cochain complex $\big(C^\bu(\cU,\cA),\delta\big)$, called 
the {\it complex of \v Cech cochains} relative to the covering $\cU$.

\claim{(5.3) Definition} The \v Cech cohomology group of $\cA$ relative
to $\cU$ is
$$\check H^q(\cU,\cA)=H^q\big(C^\bu(\cU,\cA)\big).$$
\endclaim

Formula (5.2) shows that the set of \v Cech $0$-cocycles is the set of
families $(c_\alpha)\in\prod\cA(U_\alpha)$ such that $c_\beta=c_\alpha$
on $U_\alpha\cap U_\beta$. Such a family defines in a unique way a global
section $f\in\cA(X)$ with $f_{\restriction U_\alpha}=c_\alpha$. Hence
$$\check H^0(\cU,\cA)=\cA(X).\leqno(5.4)$$
Now, let $\cV=(V_\beta)_{\beta\in J}$ be another open covering of $X$
that is finer than $\cU$~; this means that there exists a map $\rho:J\to I$
such that $V_\beta\subset U_{\rho(\beta)}$ for every $\beta\in J$. Then
we can define a morphism $\rho^\bu:C^\bu(\cU,\cA)\longrightarrow C^\bu(\cV,\cA)$ by
$$(\rho^q c)_{\beta_0\ldots\beta_q}=c_{\rho(\beta_0)\ldots\rho(\beta_q)\;
\restriction V_{\beta_0\ldots\beta_q}}~;\leqno(5.5)$$
the commutation property $\delta\rho^\bu=\rho^\bu\delta$ is immediate.
If $\rho':J\to I$ is another refinement map such that $V_\beta\subset
U_{\rho'(\beta)}$ for all $\beta$, the morphisms $\rho^\bu$, 
$\rho^{\prime\bu}$ are homotopic. To see this, we define a map 
$h^q:C^q(\cU,\cA)\longrightarrow C^{q-1}(\cV,\cA)$ by
$$(h^q c)_{\beta_0\ldots\beta_{q-1}}=\sum_{0\le j\le q-1}(-1)^j
c_{\rho(\beta_0)\ldots\rho(\beta_j)\rho'(\beta_j)\ldots\rho'(\beta_{q-1})\;
\restriction V_{\beta_0\ldots\beta_{q-1}}}.$$
The homotopy identity $\delta^{q-1}\circ h^q+h^{q+1}\circ\delta^q=
\rho^{\prime q}-\rho^q$ is easy to verify. Hence
$\rho^\bu$ and $\rho^{\prime\bu}$ induce a map depending only on 
$\cU$, $\cV$~:
$$H^q(\rho^\bu)=H^q(\rho^{\prime\bu})~:~~\check H^q(\cU,\cA)\longrightarrow
\check H^q(\cV,\cA).\leqno(5.6)$$

Now, we want to define a {\it direct limit} $\check H^q(X,\cA)$
of the groups $\check H^q(\cU,\cA)$ by means of the refinement mappings 
$(5.6)$. In order to avoid set theoretic difficulties, the coverings
used in this definition will be considered as subsets of the power set
$\cP(X)$, so that the collection of all coverings becomes actually a set.

\claim{(5.7) Definition} The \v Cech cohomology group
$\check H^q(X,\cA)$ is the direct limit
$$\check H^q(X,\cA)=\lim_{\displaystyle\,\longrightarrow\atop\scriptstyle\cU}
~~\check H^q(\cU,\cA)$$
when $\cU$ runs over the collection of all open coverings of $X$.
Explicitly, this means that the elements of $\check H^q(X,\cA)$ are the
equivalence classes in the disjoint union of the groups $\check 
H^q(\cU,\cA)$, with an element in $\check H^q(\cU,\cA)$ and another in 
$\check H^q(\cV,\cA)$ identified if their images in $\check H^q(\cW,\cA)$
coincide for some refinement $\cW$ of the coverings $\cU$ and $\cV$.
\endclaim

If $\varphi:\cA\to\cB$ is a sheaf morphism, we have an obvious induced 
morphism $\varphi^\bu:C^\bu(\cU,\cA)\longrightarrow C^\bu(\cU,\cB)$, and therefore
we find a morphism
$$H^q(\varphi^\bu):\check H^q(\cU,\cA)\longrightarrow\check H^q(\cU,\cB).$$
Let $0\to\cA\to\cB\to\cC\to 0$ be an exact sequence of sheaves. We have an
exact sequence of groups
$$0\longrightarrow C^q(\cU,\cA)\longrightarrow C^q(\cU,\cB)\longrightarrow C^q(\cU,\cC),\leqno(5.8)$$
but in general the last map is not surjective, because every section in
$\cC(U_{\alpha_0,\ldots,\alpha_q})$ need not have a lifting in 
$\cB(U_{\alpha_0,\ldots,\alpha_q})$. The image of $C^\bu(\cU,\cB)$ in
$C^\bu(\cU,\cC)$ will be denoted $C^\bu_\cB(\cU,\cC)$ and called the
complex of {\it liftable cochains} of $\cC$ in $\cB$. By construction, 
the sequence
$$0\longrightarrow C^q(\cU,\cA)\longrightarrow C^q(\cU,\cB)\longrightarrow C^q_\cB(\cU,\cC)\longrightarrow 0\leqno(5.9)$$
is exact, thus we get a corresponding long exact sequence of cohomology
$$\check H^q(\cU,\cA)\longrightarrow\check H^q(\cU,\cB)\longrightarrow\check H^q_\cB(\cU,\cC)
\longrightarrow\check H^{q+1}(\cU,\cA)\longrightarrow\cdots.\leqno(5.10)$$
If $\cA$ is flabby, Prop.\ 4.3 shows that we have
$C^q_\cB(\cU,\cC)=C^q(\cU,\cC)$, hence $\check H^q_\cB(\cU,\cC)=
\check H^q(\cU,\cC)$.

\claim{(5.11) Proposition} Let $\cA$ be a sheaf on $X$.
Assume that either
\medskip
\item{\rm a)} $\cA$ is flabby, or~$:$
\medskip
\item{\rm b)} $X$ is paracompact and $\cA$ is a sheaf of modules over a 
soft sheaf of rings $\cR$ on $X$.
\medskip
\noindent Then $\check H^q(\cU,\cA)=0$ for every $q\ge 1$ and every open 
covering $\cU=(U_\alpha)_{\alpha\in I}$ of~$X$.
\endclaim

\proof{Proof} b) Let $(\psi_\alpha)_{\alpha\in I}$ be a partition of unity in $\cR$
subordinate to $\cU$ (Prop.\ 4.17). We define a map 
$h^q:C^q(\cU,\cA)\longrightarrow C^{q-1}(\cU,\cA)$ by
$$(h^qc)_{\alpha_0\ldots\alpha_{q-1}}=\sum_{\nu\in I}\psi_\nu\,
c_{\nu\alpha_0\ldots\alpha_{q-1}}\leqno(5.12)$$
where $\psi_\nu\,c_{\nu\alpha_0\ldots\alpha_{q-1}}$ is extended by $0$ on
$U_{\alpha_0\ldots\alpha_{q-1}}\cap\complement U_\nu$. It is clear that
$$(\delta^{q-1}h^qc)_{\alpha_0\ldots\alpha_q}=\sum_{\nu\in I}
\psi_\nu\big(c_{\alpha_0\ldots\alpha_q}-(\delta^qc)_{\nu\alpha_0\ldots\alpha_q}
\big),$$
i.e.\ $\delta^{q-1}h^q+h^{q+1}\delta^q=\Id$. Hence $\delta^q c=0$ implies
$\delta^{q-1}h^qc=c$ if $q\ge1$.
\medskip
\noindent{a)} First we show that the result is true for the sheaf $\cA^{[0]}$.
One can find a family
of sets $L_\nu\subset U_\nu$ such that $(L_\nu)$ is a 
partition of $X$. If $\psi_\nu$ is the characteristic func\-tion of $L_\nu$, 
Formula (5.12) makes sense for any cochain $c\in C^q(\cU,\cA^{[0]})$ 
because $\cA^{[0]}$ is a module over the ring $\bZ^{[0]}$ of germs of
arbitrary functions $X\to\bZ$. Hence $\check H^q(\cU,\cA^{[0]})=0$
for $q\ge 1$. We shall prove this property for all flabby sheaves by 
induction on $q$. Consider the exact sequence
$$0\longrightarrow\cA\longrightarrow\cA^{[0]}\longrightarrow\cC\longrightarrow 0$$
where $\cC=\cA^{[0]}/\cA$. By the remark after (5.10), we have exact
sequences
$$\eqalign{
&\cA^{[0]}(X)\longrightarrow\cC(X)\longrightarrow\check H^1(\cU,\cA)\longrightarrow\check H^1(\cU,\cA^{[0]})=0,\cr
&\check H^q(\cU,\cC)\longrightarrow\check H^{q+1}(\cU,\cA)\longrightarrow\check H^{q+1}(\cU,\cA^{[0]})
=0.\cr}$$
Then $\cA^{[0]}(X)\longrightarrow\cC(X)$ is surjective by Prop.\ 4.3, thus
$\check H^1(\cU,\cA)=0$. By 4.3 again, $\cC$ is flabby; the induction
hypothesis $\check H^q(\cU,\cC)=0$ implies that
$\check H^{q+1}(\cU,\cA)=0$.\qed
\endproof

\subsection{5.B}{Leray's Theorem for Acyclic Coverings}
We first show the existence of a natural morphism from \v Cech cohomology to
ordinary cohomology. Let $\cU=(U_\alpha)_{\alpha\in I}$ be a covering of
$X$. Select a map $\lambda:X\to I$ such that $x\in U_{\lambda(x)}$ for
every $x\in X$. To every cochain $c\in C^q(\cU,\cA)$ we associate the
section $\lambda^qc=f\in\cA^{[q]}(X)$ such that
$$f(x_0,\ldots,x_q)=c_{\lambda(x_0)\ldots\lambda(x_q)}(x_q)\in\cA_{x_q}~;
\leqno(5.13)$$
note that the right hand side is well defined as soon as
$$x_0\in X,~~~x_1\in U_{\lambda(x_0)},~~\ldots~,~~~x_q\in 
U_{\lambda(x_0)\ldots\lambda(x_{q-1})}.$$
A comparison of (2.5) and (5.13) immediately shows that the section of
$\cA^{[q+1]}(X)$ associated to $\delta^qc$ is
$$\sum_{0\le j\le q+1}(-1)^j\,c_{\lambda(x_0)\ldots\wh{\lambda(x_j)}\ldots
\lambda(x_{q+1})}(x_{q+1})=(d^qf)(x_0,\ldots,x_{q+1}).$$
In this way we get a morphism of complexes
$\lambda^\bu:C^\bu(\cU,\cA)\longrightarrow\cA^{[\bu]}(X)$. There is a corresponding
morphism
$$H^q(\lambda^\bu):\check H^q(\cU,\cA)\longrightarrow H^q(X,\cA).\leqno(5.14)$$
If $\cV=(V_\beta)_{\beta\in J}$ is a refinement of $\cU$ such that
$V_\beta\subset U_{\rho(\beta)}$ and $x\in V_{\mu(x)}$ for all
$x,\beta$, we get a commutative diagram
$$\eqalign{
\check H^q(&\cU,\cA)\buildo H^q(\rho^\bu)\over
{\relbar\joinrel\relbar\joinrel\relbar\joinrel\longrightarrow}\check H^q(\cV,\cA)\cr
{\scriptstyle H^q(\lambda^\bu)}
&\searrow\qquad\qquad\qquad\swarrow{\scriptstyle H^q(\mu^\bu)}\cr
&\qquad H^q(X,\cA)\cr}$$
with $\lambda=\rho\circ\mu$.  In particular, (5.6) shows that the map
$H^q(\lambda^\bu)$ in (5.14) does not depend on the choice of
$\lambda$~: if $\lambda'$ is another choice, then $H^q(\lambda^\bu)$ and
$H^q(\lambda^{\prime\bu})$ can be both factorized through the group 
$\check H^q(\cV,\cA)$ with $V_x=U_{\lambda(x)}\cap U_{\lambda'(x)}$
and $\mu=\Id_X$.
By the universal property of direct limits, we get an induced morphism
$$\check H^q(X,\cA)\longrightarrow H^q(X,\cA).\leqno(5.15)$$
Let $0\to\cA\to\cB\to\cC\to 0$ be an exact sequence of sheaves. There is a
commutative diagram
$$\cmalign{
0&~\longrightarrow~ C^\bu(\cU,\cA)&~\longrightarrow~ C^\bu(\cU,\cB)&~\longrightarrow~ C^\bu_\cB(\cU,\cC)&~\longrightarrow~ 0\cr
\noalign{\vskip4pt}
&\qquad\quad~\big\downarrow&\qquad\quad~\big\downarrow
&\qquad\quad~\big\downarrow&\cr
\noalign{\vskip4pt}
0&~\longrightarrow~\cA^{[\bu]}(X)&~\longrightarrow~\cB^{[\bu]}(X)&~\longrightarrow~\cC^{[\bu]}(X)&~\longrightarrow~ 0\cr
\noalign{\vskip4pt}}$$
where the vertical arrows are given by the morphisms $\lambda^\bu$. We
obtain therefore a commutative diagram
$$\cmalign{
\noalign{\vskip4pt}
&\check H^q(\cU,\cA)&~\longrightarrow~\check H^q(\cU,\cB)&~\longrightarrow~\check
H^q_\cB(\cU,\cC)&~\longrightarrow~\check H^{q+1}(\cU,\cA)&~\longrightarrow~\check H^{q+1}(\cU,\cB)\cr
\noalign{\vskip4pt}
&\quad\big\downarrow&\qquad\quad~~\big\downarrow&\qquad\quad~~
\big\downarrow&\qquad\quad~~~\big\downarrow&\qquad\quad~~~\big\downarrow\cr
\noalign{\vskip4pt}
&H^q(X,\cA)&~\longrightarrow~ H^q(X,\cB)&~\longrightarrow~ H^q(X,\cC)&~\longrightarrow~
H^{q+1}(X,\cA)&~\longrightarrow~ H^{q+1}(X,\cB).\cr}\leqno(5.16)$$

\claim{(5.17) Theorem {\rm(Leray)}} Assume that
$$H^s(U_{\alpha_0\ldots\alpha_t},\cA)=0$$
for all indices $\alpha_0,\ldots,\alpha_t$ and $s\ge 1$. Then {\rm(5.14)}
gives an isomorphism $\check H^q(\cU,\cA)\simeq H^q(X,\cA)$.
\endclaim

We say that the covering $\cU$ is {\it acyclic} (with respect to $\cA$) if
the hypo\-thesis of Th.~5.17 is satisfied. Leray's theorem asserts
that the cohomology groups of $\cA$ on $X$ can be computed by means of an
arbitrary acyclic covering (if such a covering exists), without using the
direct limit procedure.

\proof{Proof} By induction on $q$, the result being obvious for $q=0$. 
Consider the exact sequence $0\to\cA\to\cB\to\cC\to 0$ with
$\cB=\cA^{[0]}$ and $\cC=\cA^{[0]}/\cA$.  As $\cB$ is acyclic, the
hypothesis on $\cA$ and the long exact sequence of cohomology imply
$H^s(U_{\alpha_0\ldots\alpha_t},\cC)=0$ for $s\ge 1$, $t\ge 0$. 
Moreover $C^\bu_\cB(\cU,\cC)=C^\bu(\cU,\cC)$ thanks to Cor.\ 3.6.
The induction hypothesis in degree $q$ and diagram (5.16) give
$$\cmalign{
&\check H^q(\cU,\cB)&~\longrightarrow~\check H^q(\cU,\cC)&~\longrightarrow~\check H^{q+1}(\cU,\cA)&~\longrightarrow~ 0\cr
\noalign{\vskip4pt}
&\quad~~\big\downarrow\simeq&\qquad\quad~~\big\downarrow\simeq
&\qquad\quad~~~\big\downarrow&\cr
\noalign{\vskip4pt}
&H^q(X,\cB)&~\longrightarrow~ H^q(X,\cC)&~\longrightarrow~ H^{q+1}(X,\cA)&~\longrightarrow~ 0,\cr}$$
hence $\check H^{q+1}(\cU,\cA)\longrightarrow H^{q+1}(X,\cA)$ is also an
isomorphism.\qed
\endproof

\claim{(5.18) Remark} \rm The morphism 
$H^1(\lambda^\bu):\check H^1(\cU,\cA)\longrightarrow
H^1(X,\cA)$ is always injective. Indeed, we have a commutative diagram 
$$\cmalign{
\noalign{\vskip4pt}
&\check H^0(\cU,\cB)&~\longrightarrow~\check H^0_\cB(\cU,\cC)&~\longrightarrow~\check H^1(\cU,\cA)
&~\longrightarrow~ 0\cr
\noalign{\vskip4pt}
&\quad~~\big\downarrow =&\qquad\qquad~\hookdown
&\qquad\qquad\big\downarrow&\cr
\noalign{\vskip4pt}
&H^0(X,\cB)&~\longrightarrow~ H^0(X,\cC)&~\longrightarrow~ H^1(X,\cA)&~\longrightarrow~ 0,\cr}$$
where $\check H^0_\cB(\cU,\cC)$ is the subspace of $\cC(X)=H^0(X,\cC)$
consisting of sections which can be lifted in $\cB$ over each $U_\alpha$.
As a consequence, the refinement mappings
$$H^1(\rho^\bu):\check H^1(\cU,\cA)\longrightarrow\check H^1(\cV,\cA)$$
are also injective.\qed
\endclaim

\subsection{5.C}{\v Cech Cohomology on Paracompact Spaces}
We will prove here that \v Cech cohomology theory coincides with 
the ordinary one on paracompact spaces.

\claim{(5.19) Proposition} Assume that $X$ is paracompact. If
$$0\longrightarrow\cA\longrightarrow\cB\longrightarrow\cC\longrightarrow 0$$
is an exact sequence of sheaves, there is an exact sequence
$$\check H^q(X,\cA)\longrightarrow \check H^q(X,\cB)\longrightarrow \check H^q(X,\cC)\longrightarrow \check
H^{q+1}(X,\cA)\longrightarrow\cdots$$
which is the direct limit of the exact sequences {\rm(5.10)} over all
coverings~$\cU$.
\endclaim

\proof{Proof} We have to show that the natural map
$$\lim_{\displaystyle\longrightarrow}~~\check H^q_\cB(\cU,\cC)\longrightarrow
\lim_{\displaystyle\longrightarrow}~~\check H^q(\cU,\cC)$$
is an isomorphism. This follows easily from the following lemma,
which says essentially that every cochain in $\cC$ becomes liftable
in $\cB$ after a refinement of the covering.
\endproof

\claim{(5.20) Lifting lemma} Let $\cU=(U_\alpha)_{\alpha\in I}$
be an open covering of $X$ and $c\in C^q(\cU,\cC)$.  If $X$ is paracompact,
there exists a finer covering $\cV=(V_\beta)_{\beta\in J}$ and a
refinement map $\rho:J\to I$ such that $\rho^q c\in
C^q_\cB(\cV,\cC)$.
\endclaim

\proof{Proof} Since $\cU$ admits a locally finite refinement, we may assume that
$\cU$ itself is locally finite. There exists an open covering 
$\cW=(W_\alpha)_{\alpha\in I}$ of $X$ such that $\ol W_\alpha\subset U_\alpha$.
For every point $x\in X$, we can select an open neighborhood $V_x$
of $x$ with the following properties:
\medskip
\noindent{a)} if $x\in W_\alpha$, then $V_x\subset W_\alpha$~;
\medskip
\noindent{b)} if $x\in U_\alpha$ or if $V_x\cap W_\alpha\ne\emptyset$, 
then $V_x\subset U_\alpha$~;
\medskip
\noindent{c)}$\,$ if $x\in U_{\alpha_0\ldots\alpha_q}$, then 
$c_{\alpha_0\ldots\alpha_q}\in C^q(U_{\alpha_0\ldots\alpha_q},\cC)$
admits a lifting in $\cB(V_x)$.
\medskip
\noindent Indeed, a) (resp.\ c)) can be achieved because $x$ belongs to only 
finitely many sets $W_\alpha$ (resp.\ $U_\alpha$), and so only finitely many sections
of $\cC$ have to be lifted in $\cB$. b) can be achieved because $x$ 
has a neighborhood $V'_x$ that meets only finitely many sets $U_\alpha$~; 
then we take
$$V_x\subset V'_x\cap\bigcap_{U_\alpha\ni x}U_\alpha\cap
\bigcap_{U_\alpha\not\ni x}(V'_x\ssm\ol W_\alpha).$$
Choose $\rho:X\to I$ such that $x\in W_{\rho(x)}$ for every $x$. Then 
a) implies $V_x\subset W_{\rho(x)}$, so $\cV=(V_x)_{x\in X}$ is finer than 
$\cU$, and $\rho$ defines a refinement map. 
If $V_{x_0\ldots x_q}\ne\emptyset$, we have
$$V_{x_0}\cap W_{\rho(x_j)}\supset V_{x_0}\cap V_{x_j}\ne\emptyset~~~
\hbox{\rm for}~~0\le j\le q,$$
thus $V_{x_0}\subset U_{\rho(x_0)\ldots\rho(x_q)}$ by b).
Now, c) implies that the section $c_{\rho(x_0)\ldots\rho(x_q)}$ admits a
lifting in $\cB(V_{x_0})$, and in particular in $\cB(V_{x_0\ldots x_q})$.
Therefore $\rho^qc$ is liftable in $\cB$.\qed
\endproof

\claim{(5.21) Theorem} If $X$ is a paracompact topological space,
the canonical morphism
\hbox{$\check H^q(X,\cA)\simeq H^q(X,\cA)$} is an isomorphism.
\endclaim

\proof{Proof} Argue by induction on $q$ as in Leray's theorem, with the \v Cech
coho\-mology exact sequence over $\cU$ replaced by its direct limit
in~(5.16).\qed
\endproof

In the next chapters, we will be concerned only by paracompact spaces,
and most often in fact by manifolds that are either compact or countable
at infinity. In these cases, we will not distinguish $H^q(X,\cA)$ and 
$\check H^q(X,\cA)$.

\subsection{5.D}{Alternate \v Cech Cochains}
For explicit calculations, it is sometimes useful to consider
a slightly modified \v Cech complex which has the advantage of producing
much smaller cochain groups. If $\cA$ is a
sheaf and $\cU=(U_\alpha)_{\alpha\in I}$ an open covering of $X$,
we let $AC^q(\cU,\cA)\subset C^q(\cU,\cA)$
be the subgroup of {\it alternate \v Cech cochains}, consisting
of \v Cech cochains $c=(c_{\alpha_0\ldots\alpha_q})$ such that
$$\left\{\eqalign{
c_{\alpha_0\ldots\alpha_q}&=0\phantom{(\sigma)\,c_{\alpha_0\ldots\alpha_q},}
~~~\hbox{\rm if}~~\alpha_i=\alpha_j,~~i\ne j,\cr
c_{\alpha_{\sigma(0)}\ldots\alpha_{\sigma(q)}}&=\varepsilon(\sigma)
\,c_{\alpha_0\ldots\alpha_q}\cr}\right.\leqno(5.22)$$
for any permutation $\sigma$ of $\{1,\ldots,q\}$ of signature
$\varepsilon(\sigma)$.
Then the \v Cech differential (5.1) of an alternate cochain is still
alternate, so $AC^\bu(\cU,\cA)$ is a subcomplex of $C^\bu(\cU,\cA)$.
We are going to show that the inclusion induces an isomorphism in
cohomology:
$$H^q\big(AC^\bu(\cU,\cA)\big)\simeq H^q\big(C^\bu(\cU,\cA)\big)=
\check H^q(\cU,\cA).\leqno(5.23)$$
Select a total ordering on the index set $I$. For each such ordering,
we can define a projection $\pi^q:C^q(\cU,\cA)\longrightarrow AC^q(\cU,\cA)\subset
C^q(\cU,\cA)$ by
$$c\longmapsto \hbox{\rm alternate}~\wt c~~\hbox{\rm such that}~
\wt c_{\alpha_0\ldots\alpha_q}=c_{\alpha_0\ldots\alpha_q}~~\hbox{\rm whenever}~~
\alpha_0<\ldots<\alpha_q.$$
As $\pi^\bu$ is a morphism of complexes, it is enough to verify that $\pi^\bu$ 
is homotopic to the identity on $C^\bu(\cU,\cA)$. For a given multi-index
$\alpha=(\alpha_0,\ldots,\alpha_q)$, which may contain repeated indices, there is 
a unique permutation $\big(m(0),\ldots,m(q)\big)$ of $(0,\ldots,q)$ such that 
$$\alpha_{m(0)}\le\ldots\le\alpha_{m(q)}~~~~\hbox{\rm and}~~m(l)<m(l+1)~~
\hbox{\rm whenever}~~\alpha_{m(l)}=\alpha_{m(l+1)}.$$
For $p\le q$, we let $\varepsilon(\alpha,p)$ be the sign of the permutation
$$(0,\ldots,q)\longmapsto\big(m(0),\ldots,m(p-1),0,1,\ldots,\wh{m(0)},\ldots,\wh{m(p-1)},\ldots,q\big)$$
if the elements $\alpha_{m(0)},\ldots,\alpha_{m(p)}$ are all distinct, and
$\varepsilon(\alpha,p)=0$ otherwise. Finally, we set $h^q=0$ for $q\le 0$ and 
$$(h^qc)_{\alpha_0\ldots\alpha_{q-1}}=
\sum_{0\le p\le q-1}(-1)^p\varepsilon(\alpha,p)\,c_{\alpha_{m(0)}\ldots
\alpha_{m(p)}\alpha_0\alpha_1\ldots\wh{\alpha_{m(0)}}\ldots\wh{\alpha_{m(p-1)}}
\ldots\alpha_{q-1}}$$
for $q\ge 1$~; observe that the index $\alpha_{m(p)}$ is repeated twice in 
the right hand side. A rather tedious calculation left to the reader shows that
$$(\delta^{q-1}h^qc+h^{q+1}\delta^qc)_{\alpha_0\ldots\alpha_q}=
c_{\alpha_0\ldots\alpha_q}
-\varepsilon(\alpha,q)\,c_{\alpha_{m(0)}\ldots\alpha_{m(q)}}
=(c-\pi^qc)_{\alpha_0\ldots\alpha_q}.$$
An interesting consequence of the isomorphism (5.23) is the following:

\claim{(5.24) Proposition} Let $\cA$ be a sheaf on a paracompact space
$X$.  If $X$ has arbitrarily fine open coverings or at least one
acyclic open covering $\cU=(U_\alpha)$ such that more than $n+1$
distinct sets $U_{\alpha_0},\ldots,U_{\alpha_n}$ have empty intersection, then
$H^q(X,\cA)=0$ for $q>n$.
\endclaim

\proof{Proof} In fact, we have $AC^q(\cU,\cA)=0$ for $q>n$.\qed
\endproof

\section{6}{The De Rham-Weil Isomorphism Theorem}
In \S~3 we defined cohomology groups by means of the simplicial flabby
resolution. We show here that any resolution by acyclic sheaves could
have been used instead.
Let $(\cL^{\bu},d)$ be a resolution of a sheaf $\cA$. We assume in
addition that all $\cL^q$ are acyclic on~$X$, i.e.\
$H^s(X,\cL^q)=0$ for all $q\ge 0$ and $s\ge 1$.
Set $\cZ^q=\ker d^q$. Then $\cZ^0=\cA$ and for every $q\ge 1$ we get a
short exact sequence
$$0\longrightarrow\cZ^{q-1}\longrightarrow\cL^{q-1}\buildo d^{q-1}\over\longrightarrow\cZ^q\longrightarrow 0.$$
Theorem 3.5 yields an exact sequence
$$H^s(X,\cL^{q-1})\buildo d^{q-1}\over\longrightarrow H^s(X,\cZ^q) \buildo
\partial^{s,q}\over\longrightarrow H^{s+1}(X,\cZ^{q-1})\to H^{s+1}(X,\cL^{q-1})
=0.\leqno(6.1)$$
If $s\ge 1$, the first group is also zero and we get an isomorphism
$$\partial^{s,q}:H^s(X,\cZ^q)\buildo\simeq\over\longrightarrow H^{s+1}(X,\cZ^{q-1}).$$
For $s=0$ we have $H^0(X,\cL^{q-1})=\cL^{q-1}(X)$ and
$H^0(X,\cZ^q)=\cZ^q(X)$ is the $q$-cocycle group of
$\cL^\bu(X)$, so the connecting map $\partial^{0,q}$ gives an isomorphism
$$H^q\big(\cL^\bu(X)\big)=\cZ^q(X)/d^{q-1}\cL^{q-1}(x)
\buildo\wt\partial^{0,q}\over\longrightarrow H^1(X,\cZ^{q-1}).$$
The composite map $\partial^{q-1,1}\circ\cdots\circ\partial^{1,q-1}\circ
\wt\partial^{0,q}$ therefore defines an isomorphism
$$\leqalignno{\quad\qquad&H^q\big(\cL^\bu(X)\big)\!\buildo\wt\partial^{0,q}
\over\longrightarrow\! H^1(X,\cZ^{q-1})\!\buildo\partial^{1,q-1}\over\longrightarrow\!\cdots\!
\buildo\partial^{q-1,1}\over\longrightarrow\! H^q(X,\cZ^0){=}H^q(X,\cA).&(6.2)\cr}$$
This isomorphism behaves functorially with respect to morphisms of 
reso\-lutions.  Our assertion means that for every sheaf morphism
$\varphi:\cA\to\cB$  and every morphism of resolutions
$\varphi^\bu:\cL^\bu\longrightarrow\cM^\bu$, there is a commutative diagram
$$\cmalign{
&H^s\big(&\cL^\bu(X)\big)\hfill\longrightarrow H^s(&X,\cA)\cr
\noalign{\vskip5pt}
&&\big\downarrow H^s(\varphi^\bu)\hfill&\big\downarrow H^s(\varphi)\hfill\cr
\noalign{\vskip5pt}
&H^s\big(&\cM^\bu(X)\big)\longrightarrow H^s(&X,\cB).\cr}\leqno(6.3)$$
If $\cW^q=\ker(d^q:\cM^q\to\cM^{q+1})$, the functoriality comes
from the fact that we have commutative diagrams
$$\cmalign{
0\to&\cZ^{q-1}\hfill\to&\cL^{q-1}\hfill\to&\cZ^q\to 0~,~~~~
H^s(&X,\cZ^q)\hfill\buildo\partial^{s,q}\over\longrightarrow H^{s+1}(&X,\cZ^{q-1})\cr
\noalign{\vskip5pt}
&\big\downarrow\varphi^{q-1}\hfill&~\big\downarrow\varphi^{q-1}\hfill
&\big\downarrow\varphi^q\hfill&\big\downarrow H^s(\varphi^q)\hfill
&\big\downarrow H^{s+1}(\varphi^{q-1})\hfill\cr
\noalign{\vskip5pt}
0\to&\cW^{q-1}\to&\cM^{q-1}\to&\cW^q\to 0~,~~~~
H^s(&X,\cW^q)\buildo\partial^{s,q}\over\longrightarrow H^{s+1}(&X,\cW^{q-1}).\cr}$$

\claim{(6.4) De Rham-Weil isomorphism theorem} If $(\cL^\bu,d)$ is
a resolution of $\cA$ by sheaves $\cL^q$ which are acyclic on $X$, there
is a functorial isomorphism 
$$H^q\big(\cL^\bu(X)\big)\longrightarrow H^q(X,\cA).\eqno\square$$
\endclaim

\claim{(6.5) Example: De Rham cohomology} \rm
Let $X$ be a $n$-dimensional paracompact differential
manifold. Consider the resolution
$$0\to\bR\to\cE^0\buildo d\over\to\cE^1\to\cdots\to\cE^q\buildo d\over\to
\cE^{q+1}\to\cdots\to\cE^n\to 0$$
given by the exterior derivative $d$ acting on germs of $\cC^\infty$
differential $q$-forms (cf.\ Example~2.2). The {\it De Rham cohomology
groups} of $X$ are precisely
$$H^q_{\DR}(X,\bR)=H^q\big(\cE^\bu(X)\big).\leqno(6.6)$$
All sheaves $\cE^q$ are $\cE_X$-modules, so $\cE^q$ is acyclic by 
Cor.\ 4.19. Therefore, we get an isomorphism
$$H^q_{\DR}(X,\bR)\buildo\simeq\over\longrightarrow H^q(X,\bR)\leqno(6.7)$$
from the De Rham cohomology onto the cohomology with values in the 
constant sheaf~$\bR$.  Instead of using $\cC^\infty$ differential forms, 
one can consider the resolution of $\bR$ given by the exterior derivative
$d$ acting on currents:
$$0\to\bR\to\cD'_n\buildo d\over\to\cD'_{n-1}\to\cdots\to\cD'_{n-q}
\buildo d\over\to\cD'_{n-q-1}\to\cdots\to\cD'_0\to 0.$$
The sheaves $\cD'_q$ are also $\cE_X$-modules, hence acyclic. Thanks 
to (6.3), the inclusion $\cE^q\subset\cD_{n-q}'$ induces an isomorphism
$$H^q\big(\cE^\bu(X)\big)\simeq H^q\big(\cD'_{n-\bu}(X)\big),\leqno(6.8)$$
both groups being isomorphic to $H^q(X,\bR)$. The isomorphism between
cohomology of differential forms and singular cohomology (another
topological invariant) was first established by \cite{De Rham
1931}. The above proof follows essentially the method given by 
\cite{Weil 1952}, in a more abstract setting. As we will see, the isomorphism
$(6.7)$ can be put under a very explicit form in terms of \v Cech
cohomology. We need a simple lemma.
\endclaim

\claim{(6.9) Lemma} Let $X$ be a paracompact differentiable manifold.
There are arbitrarily fine open coverings $\cU=(U_\alpha)$ such that
all intersections $U_{\alpha_0\ldots\alpha_q}$ are diffeomorphic to
convex sets.
\endclaim

\proof{Proof} Select locally finite coverings
$\Omega'_j\subset\!\subset\Omega_j$ of $X$ by open sets diffeomorphic to
concentric euclidean balls in $\bR^n$.  Let us denote by
$\tau_{jk}$ the transition diffeomorphism from the coordinates in
$\Omega_k$ to those in $\Omega_j$.  For any point $a\in\Omega'_j$, the
function $x\mapsto|x-a|^2$ computed in terms of the coordinates of
$\Omega_j$ becomes $|\tau_{jk}(x)-\tau_{jk}(a)|^2$ on
any patch $\Omega_k\ni a$.  It is clear that these functions are
strictly convex at $a$, thus there is a euclidean ball
$B(a,\varepsilon)\subset\Omega'_j$ such that all functions are
strictly convex on $B(a,\varepsilon)\cap\Omega'_k \subset\Omega_k$ (only
a finite number of indices $k$ is involved).  Now, choose $\cU$ to be a
(locally finite) covering of $X$ by such balls 
$U_\alpha=B(a_\alpha,\varepsilon_\alpha)$ with
$U_\alpha\subset\Omega'_{\rho(\alpha)}$.  Then the
intersection $U_{\alpha_0\ldots\alpha_q}$ is defined in
$\Omega_k$, $k=\rho(\alpha_0)$, by the equations
$$|\tau_{jk}(x)-\tau_{jk}(a_{\alpha_m})|^2<\varepsilon_{\alpha_m}^2$$
where $j=\rho(\alpha_m)$, $0\le m\le q$. Hence the intersection
is convex in the open coordinate chart $\Omega_{\rho(\alpha_0)}$.\qed
\endproof

Let $\Omega$ be an open subset of $\bR^n$ which is starshaped with 
respect to the origin. Then the De Rham
complex $\bR\longrightarrow\cE^\bu(\Omega)$ is acyclic: indeed, Poincar\'e's lemma
yields a homotopy operator $k^q:\cE^q(\Omega)\longrightarrow\cE^{q-1}(\Omega)$ such that
$$\eqalign{
&k^qf_x(\xi_1,\ldots,\xi_{q-1})=\int_0^1t^{q-1}\,f_{tx}(x,\xi_1,\ldots,\xi_{q-1})\,dt,
~~~x\in\Omega,~~\xi_j\in\bR^n,\cr
&k^0f=f(0)\in\bR~~~\hbox{\rm for}~~f\in\cE^0(\Omega).\cr}$$
Hence $H^q_{\DR}(\Omega,\bR)=0$ for $q\ge 1$. Now, consider the resolution
$\cE^\bu$ of the constant sheaf $\bR$ on $X$, and apply the proof of the 
De Rham-Weil isomorphism theorem to \v Cech cohomology groups over a
covering $\cU$ chosen as in Lemma 6.9. Since the intersections 
$U_{\alpha_0\ldots\alpha_s}$ are convex, all \v Cech cochains in 
$C^s(\cU,\cZ^q)$ are liftable in $\cE^{q-1}$ by means of $k^q$.
Hence for all $s=1,\ldots,q$ we have isomorphisms
$\partial^{s,q-s}:\check H^s(\cU,\cZ^{q-s})\longrightarrow\check H^{s+1}(\cU,\cZ^{q-s-1})$
for $s\ge 1$ and we get a resulting isomorphism 
$$\partial^{q-1,1}\circ\cdots\circ\partial^{1,q-1}\circ
\wt\partial^{0,q}:H^q_{\DR}(X,\bR)\buildo\simeq\over\longrightarrow\check H^q(\cU,\bR)$$
We are going to compute the connecting homomorphisms $\partial^{s,q-s}$
and their inverses explicitly. 

Let $c$ in $C^s(\cU,\cZ^{q-s})$ such that
$\delta^sc=0$. As $c_{\alpha_0\ldots\alpha_s}$ is $d$-closed, we can write 
$c=d(k^{q-s}c)$ where the cochain $k^{q-s}c\in C^s(\cU,\cE^{q-s-1})$
is defined as the family of sections
$k^{q-s}c_{\alpha_0\ldots\alpha_s}\in\cE^{q-s-1}(U_{\alpha_0\ldots\alpha_s})$.
Then $d(\delta^sk^{q-s}c)=\delta^s(dk^{q-s}c)=\delta^sc=0$ and
$$\partial^{s,q-s}\{c\}=\{\delta^sk^{q-s}c\}\in\check 
H^{s+1}(\cU,\cZ^{q-s-1}).$$
The isomorphism
$H^q_{\DR}(X,\bR)\buildo\simeq\over\longrightarrow\check H^q(\cU,\bR)$ is thus defined as
follows: to the cohomology class $\{f\}$ of a closed $q$-form 
$f\in\cE^q(X)$, we associate the cocycle $(c^0_\alpha)=
(f_{\restriction U_\alpha})\in C^0(\cU,\cZ^q)$, then the cocycle
$$c^1_{\alpha\beta}=k^qc^0_\beta-k^qc^0_\alpha\in C^1(\cU,\cZ^{q-1}),$$
and by induction cocycles
$(c^s_{\alpha_0\ldots\alpha_s})\in C^s(\cU,\cZ^{q-s})$ given by
$$c^{s+1}_{\alpha_0\ldots\alpha_{s+1}}=\sum_{0\le j\le s+1}(-1)^j\,
k^{q-s}c^s_{\alpha_0\ldots\wh{\alpha_j}\ldots\alpha_{s+1}}~~~~
\hbox{\rm on}~~U_{\alpha_0\ldots\alpha_{s+1}}.\leqno(6.10)$$
The image of $\{f\}$ in $\check H^q(\cU,\bR)$ is the class of the $q$-cocycle
$(c^q_{\alpha_0\ldots\alpha_q})$ in $C^q(\cU,\bR)$.

Conversely, let $(\psi_\alpha)$ be a $\cC^\infty$ partition of unity subordinate to
$\cU$. Any \v Cech cocycle $c\in C^{s+1}(\cU,\cZ^{q-s-1})$ can be written
$c=\delta^s\gamma$ with $\gamma\in C^s(\cU,\cE^{q-s-1})$ given by
$$\gamma_{\alpha_0\ldots\alpha_s}=\sum_{\nu\in I}\psi_\nu\,
c_{\nu\alpha_0\ldots\alpha_s},$$
(cf.\ Prop.\ 5.11~b)), thus $\{c'\}=(\partial^{s,q-s})^{-1}\{c\}$
can be represented by the cochain $c'=d\gamma\in C^s(\cU,\cZ^{q-s})$ such that
$$c'_{\alpha_0\ldots\alpha_s}=\sum_{\nu\in I}d\psi_\nu\wedge
c_{\nu\alpha_0\ldots\alpha_s}=(-1)^{q-s-1}\sum_{\nu\in I}
c_{\nu\alpha_0\ldots\alpha_s}\wedge d\psi_\nu.$$
For a reason that will become apparent later, we shall in fact modify the
sign of our isomorphism $\partial^{s,q-s}$ by the factor $(-1)^{q-s-1}$.
Starting from a class $\{c\}\in\check H^q(\cU,\bR)$, we obtain inductively
$\{b\}\in\check H^0(\cU,\cZ^q)$ such that
$$b_{\alpha_0}=\sum_{\nu_0,\ldots,\nu_{q-1}}c_{\nu_0\ldots\nu_{q-1}\alpha_0}\,
d\psi_{\nu_0}\wedge\ldots\wedge d\psi_{\nu_{q-1}}~~~~\hbox{\rm on}~~U_{\alpha_0},
\leqno(6.11)$$
corresponding to $\{f\}\in H^q_{\DR}(X,\bR)$ given by the explicit formula
$$f=\sum_{\nu_q}\psi_{\nu_q}b_{\nu_q}=
\sum_{\nu_0,\ldots,\nu_q}c_{\nu_0\ldots\nu_q}\,\psi_{\nu_q}
d\psi_{\nu_0}\wedge\ldots\wedge d\psi_{\nu_{q-1}}.\leqno(6.12)$$
The choice of sign corresponds to (6.2) multiplied by $(-1)^{q(q-1)/2}$.

\claim{(6.13) Example: Dolbeault cohomology groups} \rm
Let $X$ be a $\bC$-analytic manifold of dimension $n$, and let 
$\cE^{p,q}$ be the sheaf of germs of $\cC^\infty$ differential forms of type $(p,q)$
with complex values. For every $p=0,1,\ldots,n$, the Dolbeault-Grothendieck
Lemma I-2.9 shows that $(\cE^{p,\bu},d'')$ is a resolution of 
the sheaf $\Omega^p_X$ of germs of holomorphic forms of degree $p$ on $X$.
The {\it Dolbeault cohomology groups} of $X$ already considered in chapter~1
can be defined by
$$H^{p,q}(X,\bC)=H^q\big(\cE^{p,\bu}(X)\big).\leqno(6.14)$$
The sheaves $\cE^{p,q}$ are acyclic, so we get the {\it Dolbeault 
isomorphism theorem}, originally proved in \cite{Dolbeault 1953}, 
which relates $d''$-cohomology and sheaf cohomology:
$$H^{p,q}(X,\bC)\buildo\simeq\over\longrightarrow H^q(X,\Omega^p_X).
\leqno(6.15)$$
The case $p=0$ is especially interesting:
$$H^{0,q}(X,\bC)\simeq H^q(X,\cO_X).\leqno(6.16)$$
As in the case of De Rham cohomology, there is an inclusion
$\cE^{p,q}\subset\cD'_{n-p,n-q}$ and the complex of currents
$(\cD'_{n-p,n-\bu},d'')$ defines also a resolution of $\Omega^p_X$.
Hence there is an isomorphism:
$$H^{p,q}(X,\bC)=H^q\big(\cE^{p,\bu}(X)\big)\simeq 
H^q\big(\cD'_{n-p,n-\bu}(X)\big).\leqno(6.17)$$
\endclaim

\section{7}{Cohomology with Supports}
As its name indicates, cohomology with supports deals with sections of
sheaves having supports in prescribed closed sets.
We first introduce what is an admissible family of supports.

\claim{(7.1) Definition} A family of supports on a topological space $X$
is a collection $\Phi$ of closed subsets of $X$ with the following
two properties:
\medskip
\item{\rm a)} If $F\,,\,F'\in\Phi$, then $F\cup F'\in\Phi~;$
\medskip
\item{\rm b)} If $F\in\Phi$ and $F'\subset F$ is closed, then $F'\in\Phi.$
\endclaim

\claim{(7.2) Example} \rm Let $S$ be an arbitrary subset of $X$. Then the family 
of all closed subsets of $X$ contained in $S$ is a family of supports.
\endclaim

\claim{(7.3) Example} \rm The collection of all compact (non necessarily
Hausdorff) subsets of $X$ is a family of supports, which will be denoted 
simply $c$ in the sequel.\qed
\endclaim

\claim{(7.4) Definition} For any sheaf $\cA$ and any family of
supports $\Phi$ on $X$, $\cA_\Phi(X)$ will denote the set of
all sections $f\in\cA(X)$ such that $\Supp\,f\in\Phi$.
\endclaim

It is clear that $\cA_\Phi(X)$ is a subgroup of $\cA(X)$. We can now
introduce cohomology groups with arbitrary supports.

\claim{(7.5) Definition} The cohomology groups of $\cA$ with supports in
$\Phi$ are
$$H^q_\Phi(X,\cA)=H^q\big(\cA^{[\bu]}_\Phi(X)\big).$$
The cohomology groups with compact supports will be denoted
$H^q_c(X,\cA)$ and the cohomology groups with supports in a subset $S$
will be denoted $H^q_S(X,\cA)$.
\endclaim

In particular $H^0_\Phi(X,\cA)=\cA_\Phi(X)$. If $0\to\cA\to\cB\to\cC\to 0$ 
is an exact sequence, there are corresponding exact sequences
$$\cmalign{
&\hfil 0~~\longrightarrow~&~~\cA^{[q]}_\Phi(X)&~\longrightarrow~~~\cB^{[q]}_\Phi(X)
&~\longrightarrow~~~\cC^{[q]}_\Phi(X)&~\longrightarrow~\cdots\cr
&&~~H_\Phi^q(X,\cA)&~\longrightarrow~ H_\Phi^q(X,\cB)&~\longrightarrow~ H_\Phi^q(X,\cC)&~\longrightarrow~
H_\Phi^{q+1}(X,\cA)~\longrightarrow~\cdots.\cr}\leqno{\displaystyle(7.6)\atop
\strut}$$
When $\cA$ is flabby, there is an exact sequence
$$0\longrightarrow\cA_\Phi(X)\longrightarrow\cB_\Phi(X)\longrightarrow\cC_\Phi(X)\longrightarrow 0\leqno(7.7)$$
and every $g\in\cC_\Phi(X)$ can be lifted to $v\in\cB_\Phi(X)$ without
enlarging the support: apply the proof of Prop.\ 4.3 to a
maximal lifting which extends $w=0$ on $W=\complement(\Supp\,g)$.
It follows that a flabby sheaf $\cA$ is $\Phi$-acyclic, i.e.\
$H^q_\Phi(X,\cA)=0$ for all $q\ge 1$. Similarly, assume that
$X$ is paracompact and that $\cA$ is soft, and suppose that $\Phi$ has the
following additional property: every set $F\in\Phi$ has a neighborhood 
$G\in\Phi$. An adaptation of the proofs of Prop.\ 4.3 and
4.13 shows that (7.7) is again exact. Therefore every soft sheaf
is also $\Phi$-acyclic in that case.

As a consequence of (7.6), any resolution $\cL^\bu$ of $\cA$
by $\Phi$-acyclic sheaves provides a canonical De Rham-Weil isomorphism
$$H^q\big(\cL^\bu_\Phi(X)\big)\longrightarrow H^q_\Phi(X,\cA).\leqno(7.8)$$

\claim{(7.9) Example: De Rham cohomology with compact support}
\rm In the special case of the De Rham resolution $\bR\longrightarrow\cE^\bu$ on
a paracompact manifold, we get an isomorphism
$$H^q_{\DR,c}(X,\bR):=H^q\big((\cD^\bu(X)\big)\buildo\simeq\over\longrightarrow
H^q_c(X,\bR),\leqno(7.10)$$
where $\cD^q(X)$ is the space of smooth differential $q$-forms with
compact support in $X$. These groups are called the {\it De Rham 
cohomology groups} of $X$ with compact support.
When $X$ is oriented, $\dim X=n$, we can also consider the
complex of compactly supported currents:
$$0\longrightarrow\cE'_n(X)\buildo d\over\longrightarrow\cE'_{n-1}(X)\longrightarrow\cdots\longrightarrow\cE'_{n-q}(X)
\buildo d\over\longrightarrow\cE'_{n-q-1}(X)\longrightarrow\cdots.$$
Note that $\cD^\bu(X)$ and $\cE'_{n-\bu}(X)$ are respectively the subgroups
of compactly supported sections in $\cE^\bu$ and $\cD'_{n-\bu}$, both of
which are acyclic resolutions~of~$\bR$. Therefore the inclusion 
$\cD^\bu(X)\subset\cE'_{n-\bu}(X)$ induces an isomorphism 
$$H^q\big(\cD^\bu(X)\big)\simeq H^q\big(\cE'_{n-\bu}(X)\big),\leqno(7.11)$$
both groups being isomorphic to $H^q_c(X,\bR)$.\qed
\endclaim

Now, we concentrate our attention on cohomology groups with compact
support. We assume until the end of this section that $X$ is a
{\it locally compact} space.

\claim{(7.12) Proposition} There is an isomorphism
$$H^q_c(X,\cA)=\lim_{\displaystyle\,\longrightarrow\atop\scriptstyle
U\subset\!\subset X}~~H^q(\ol U,\cA_U)$$
where $\cA_U$ is the sheaf of sections of $\cA$ vanishing on
$X\ssm U$ (cf.\ \S 3).
\endclaim

\proof{Proof} By definition
$$H^q_c(X,\cA)=H^q\big(\cA^{[\bu]}_c(X)\big)
=\lim_{\displaystyle\,\longrightarrow\atop\scriptstyle U\subset\!\subset X}~~
H^q\big((\cA^{[\bu]})_U(\ol U)\big)$$
since sections of $(\cA^{[\bu]})_U(\ol U)$ can be extended by $0$
on $X\ssm U$. However, $(\cA^{[\bu]})_U$ is a resolution of
$\cA_U$ and $\smash{(\cA^{[q]})_U}$ is a $\smash{\bZ^{[q]}}$-module, 
so it is acyclic on $\smash{\ol U}$. The De Rham-Weil isomorphism 
theorem implies
$$H^q\big((\cA^{[\bu]})_U(\ol U)\big)=H^q(\ol U,\cA_U)$$
and the proposition follows. The reader should take care of the fact
that $(\cA^{[q]})_U$ does not coincide in general with 
$(\cA_U)^{[q]}$.\qed
\endproof

The cohomology groups with compact support can also be defined by means
of \v Cech cohomology.

\claim{(7.13) Definition} Assume that $X$ is a separable locally compact
space. If $\cU=(U_\alpha)$ is a locally finite
covering of $X$ by relatively compact open subsets, we let
$C^q_c(\cU,\cA)$ be the subgroups of cochains such that only finitely
many coefficients $c_{\alpha_0\ldots\alpha_q}$ are non zero. The
\v Cech cohomology groups with compact support are defined by
$$\eqalign{
&\check H^q_c(\cU,\cA)=H^q\big(C^\bu_c(\cU,\cA)\big)\cr
&\check H^q_c(X,\cA)=\lim_{{\displaystyle\longrightarrow}\atop{\scriptstyle\cU}}
H^q\big(C^\bu_c(\cU,\cA)\big)\cr}$$
\endclaim

For such coverings $\cU$, Formula (5.13) yields canonical morphisms
$$H^q(\lambda^\bu)~:~~\check H^q_c(\cU,\cA)\longrightarrow H^q_c(X,\cA).\leqno(7.14)$$
Now, the lifting Lemma 5.20 is valid for cochains with compact
supports, and the same proof as the one given in \S 5 gives an
isomorphism
$$\check H^q_c(X,\cA)\simeq H^q_c(X,\cA).\leqno(7.15)$$

\section{8}{Cup Product}
Let $\cR$ be a sheaf of commutative rings and $\cA$, $\cB$ sheaves of
$\cR$-modules on a space $X$. We denote by $\cA\otimes_\cR\cB$ the sheaf 
on $X$ defined by
$$(\cA\otimes_\cR\cB)_x=\cA_x\otimes_{\cR_x}\cB_x,\leqno(8.1)$$
with the weakest topology such that the range of any section given by
$\cA(U)\otimes_{\cR(U)}\cB(U)$ is open in $\cA\otimes_\cR\cB$ for any
open set $U\subset X$. Given $f\in\cA^{[p]}_x$ and $g\in\cB^{[q]}_x$, the
{\it cup product} $f\smallsmile g\in(\cA\otimes_\cR\cB)^{[p+q]}_x$ is defined by
$$f\smallsmile g(x_0,\ldots,x_{p+q})=f(x_0,\ldots,x_p)(x_{p+q})\otimes g(x_p,\ldots,x_{p+q}).
\leqno(8.2)$$
A simple computation shows that
$$d^{p+q}(f\smallsmile g)=(d^pf)\smallsmile g+(-1)^p\,f\smallsmile(d^qg).
\leqno(8.3)$$
In particular, $f\smallsmile g$ is a cocycle if $f,g$ are cocycles, and we have
$$(f+d^{p-1}f')\smallsmile(g+d^{q-1}g')=f\smallsmile g+d^{p+q-1}
\big(f'\smallsmile g+(-1)^pf\smallsmile g'+f'\smallsmile dg'\big).$$
Consequently, there is a well defined $\cR(X)$-bilinear morphism
$$H^p(X,\cA)\times H^q(X,\cB)\longrightarrow H^{p+q}(X,\cA\otimes_\cR\cB)\leqno(8.4)$$
which maps a pair $(\{f\},\{g\})$ to $\{f\smallsmile g\}$.

Let $0\to\cB\to\cB'\to\cB''\to 0$ be an exact sequence of sheaves. Assume that 
the sequence obtained after taking the tensor product by $\cA$ is also exact:
$$0\longrightarrow\cA\otimes_\cR\cB\longrightarrow\cA\otimes_\cR\cB'\longrightarrow\cA\otimes_\cR\cB''\longrightarrow 0.$$
Then we obtain connecting homomorphisms
$$\eqalign{
&\partial^q~:~~H^q(X,\cB'')\longrightarrow H^{q+1}(X,\cB),\cr
&\partial^q~:~~H^q(X,\cA\otimes_\cR\cB'')\longrightarrow H^{q+1}(X,\cA\otimes_\cR\cB).
\cr}$$
For every $\alpha\in H^p(X,\cA)$, $\beta''\in H^q(X,\cB'')$ we have
$$\leqalignno{
\partial^{p+q}(\alpha\smallsmile\beta'')&=(-1)^p\,\alpha\smallsmile(\partial^q\beta''),
&(8.5)\cr
\partial^{p+q}(\beta''\smallsmile\alpha)&=(\partial^q\beta'')\smallsmile\alpha,
&(8.5')\cr}$$
where the second line corresponds to the tensor product of the exact sequence
by $\cA$ on the right side. These formulas are deduced from (8.3) applied to a
repre\-sentant $f\in\cA^{[p]}(X)$ of $\alpha$ and to a lifting $g'\in
\cB^{\prime[q]}(X)$ of a representative $g''$ of $\beta''$ (note that $d^pf=0$).

\claim{(8.6) Associativity and anticommutativity} Let 
$i:\cA\otimes_\cR\cB\longrightarrow\cB\otimes_\cR\cA$ be the
canonical isomorphism $s\otimes t\mapsto t\otimes s$. For all
$\alpha\in H^p(X,\cA)$, $\beta\in H^q(X,\cB)$ we have
$$\beta\smallsmile\alpha=(-1)^{pq}\,i(\alpha\smallsmile\beta).$$
If $\cC$ is another sheaf of $\cR$-modules and $\gamma\in H^r(X,\cC)$,
then
$$(\alpha\smallsmile\beta)\smallsmile\gamma=
\alpha\smallsmile(\beta\smallsmile\gamma).$$
\endclaim

\proof{Proof} The associativity property is easily seen to hold already for all
cochains
$$(f\smallsmile g)\smallsmile h=f\smallsmile(g\smallsmile h),~~~f\in\cA^{[p]}_x,~~g\in\cB^{[q]}_x,~~
h\in\cC^{[r]}_x.$$
The commutation property is obvious for $p=q=0$, and can be proved in general
by induction on $p+q$. Assume for example $q\ge 1$. Consider the exact
sequence
$$0\longrightarrow\cB\longrightarrow\cB'\longrightarrow\cB''\longrightarrow 0$$
where $\cB'=\cB^{[0]}$ and $\cB''=\cB^{[0]}/\cB$. This exact sequence splits
on each stalk (but not globally, nor even locally): a left inverse 
$\smash{\cB^{[0]}_x}\to\cB_x$ of the inclusion is given by $g\mapsto g(x)$.
Hence the sequence remains exact after taking the tensor product with $\cA$.
Now, as $\cB'$ is acyclic, the connecting homomorphism 
$H^{q-1}(X,\cB'')\longrightarrow H^q(X,\cB)$ is onto, so there is
$\beta''\in H^{q-1}(X,\cB'')$ such that $\beta=\partial^{q-1}\beta''$.
Using (8.$5'$), (8.5) and the induction hypothesis, we find
$$\eqalignno{
\beta\smallsmile\alpha&=\partial^{p+q-1}(\beta''\smallsmile\alpha)=\partial^{p+q-1}\big(
(-1)^{p(q-1)}\,i(\alpha\smallsmile\beta'')\big)\cr
&=(-1)^{p(q-1)}\,i\partial^{p+q-1}(\alpha\smallsmile\beta'')=
(-1)^{p(q-1)}(-1)^p\,i(\alpha\smallsmile\beta).&\square\cr}$$
\endproof

Theorem 8.6 shows in particular that $H^\bu(X,\cR)$ is a graded associative
and supercommutative algebra, i.e.\ $\beta\smallsmile\alpha=(-1)^{pq}\,\alpha\smallsmile\beta$
for all classes $\alpha\in H^p(X,\cR)$, $\beta\in H^q(X,\cR)$. If $\cA$ is a 
$\cR$-module, then $H^\bu(X,\cA)$ is a graded $H^\bu(X,\cR)$-module.

\claim{(8.7) Remark} \rm The cup product can also be defined for \v Cech cochains.
Given $c\in C^p(\cU,\cA)$ and $c'\in C^q(\cU,\cB)$, the cochain
$c\smallsmile c'\in C^{p+q}(\cU,\cA\otimes_\cR\cB)$ is defined by
$$(c\smallsmile c')_{\alpha_0\ldots\alpha_{p+q}}=c_{\alpha_0\ldots\alpha_p}\otimes
c'_{\alpha_p\ldots\alpha_{p+q}}~~~\hbox{\rm on}~~U_{\alpha_0\ldots\alpha_{p+q}}.$$
Straightforward calculations show that
$$\delta^{p+q}(c\smallsmile c')=(\delta^pc)\smallsmile c'+(-1)^p\,c\smallsmile(\delta^qc')$$
and that there is a commutative diagram 
$$\cmalign{
\noalign{\vskip2pt}
\check H^p(\cU,\cA)&\times\check H^q(\cU,\cB)&~\longrightarrow~
\check H^{p+q}(\cU&,\cA\otimes_\cR\cB)\cr
\noalign{\vskip4pt}
&\big\downarrow&&\big\downarrow\cr
\noalign{\vskip4pt}
H^p(X,\cA)&\times H^q(X,\cB)&~\longrightarrow~H^{p+q}(X&,\cA\otimes_\cR\cB),\cr
\noalign{\vskip2pt}
}$$
where the vertical arrows are the canonical morphisms $H^s(\lambda^\bu)$
of (5.14). Note that the commutativity already holds in fact on
cochains.
\endclaim

\claim{(8.8) Remark} \rm Let $\Phi$ and $\Psi$ be families of supports on $X$.
Then $\Phi\cap\Psi$ is again a family of supports, and Formula (8.2) 
defines a bilinear map
$$H_\Phi^p(X,\cA)\times H_\Psi^q(X,\cB)\longrightarrow H_{\Phi\cap\Psi}^{p+q}
(X,\cA\otimes_\cR\cB)\leqno(8.9)$$
on cohomology groups with supports. This follows immediately from the fact
that $\Supp(f\smallsmile g)\subset\Supp\,f\cap\Supp\,g$.
\endclaim

\claim{(8.10) Remark} \rm Assume that $X$ is a differentiable manifold.  Then the
cohomology complex $H^\bu_{\DR}(X,\bR)$ has a natural structure of
supercommutative algebra given by the wedge product of differential
forms.  We shall prove the following compatibility statement:
\medskip
\noindent{\it Let $H^q(X,\bR)\longrightarrow H^q_{\DR}(X,\bR)$ be the De
Rham-Weil isomorphism given by Formula {\rm (6.12)}.  Then the cup
product $c'\smallsmile c''$ is mapped on the wedge product $f'\wedge
f''$ of the corresponding De Rham cohomology classes.}
\medskip
\noindent By remark 8.7, we may suppose that $c',c''$ are \v Cech 
cohomology classes of respective degrees $p,q$. Formulas (6.11) and 
(6.12) give
$$\eqalign{
f'_{\restriction U_{\nu_p}}&=\sum_{\nu_0,\ldots,\nu_{p-1}}
c'_{\nu_0\ldots\nu_{p-1}\nu_p}\,d\psi_{\nu_0}\wedge\ldots\wedge 
d\psi_{\nu_{p-1}},\cr
f''&=\sum_{\nu_p,\ldots,\nu_{p+q}}
c''_{\nu_p\ldots\nu_{p+q}}\,\psi_{\nu_{p+q}}d\psi_{\nu_p}\wedge\ldots\wedge 
d\psi_{\nu_{p+q-1}}.\cr}$$
We get therefore
$$f'\wedge f''=\sum_{\nu_0,\ldots,\nu_{p+q}}
c'_{\nu_0\ldots\nu_p}\,c''_{\nu_p\ldots\nu_{p+q}}\,
\psi_{\nu_{p+q}}d\psi_{\nu_0}\wedge\ldots\wedge\psi_{\nu_{p+q-1}},$$
which is precisely the image of $c\smallsmile c'$ in the De Rham 
cohomology.\qed
\endclaim

\section{9}{Inverse Images and Cartesian Products}
\subsection{9.A}{Inverse Image of a Sheaf}
Let $F:X\to Y$ be a continuous map between topological spaces $X,Y$, and
let $\pi:\cA\to Y$ be a sheaf of abelian groups. Recall that {\it inverse image}
$F^{-1}\cA$ is defined as the sheaf-space
$$F^{-1}\cA=\cA\times_Y X=\big\{(s,x)\,;\,\pi(s)=F(x)\big\}$$
with projection $\pi'=\pr_2:F^{-1}\cA\to X$. The stalks of $F^{-1}\cA$ are
given by
$$(F^{-1}\cA)_x=\cA_{F(x)},\leqno(9.1)$$
and the sections $\sigma\in F^{-1}\cA(U)$ can be considered as continuous
mappings \hbox{$\sigma:U\to\cA$} such that $\pi\circ\sigma=F$. In particular,
any section
$s\in\cA(U)$ has a {\it pull-back}
$$F^\star s=s\circ F\in F^{-1}\cA\big(F^{-1}(U)\big).\leqno(9.2)$$
For any $v\in\cA^{[q]}_y$, we define $F^\star v\in(F^{-1}\cA)^{[q]}_x$ by
$$F^\star v(x_0,\ldots,x_q)=v\big(F(x_0),\ldots,F(x_q)\big)\in(F^{-1}\cA)_{x_q}=
\cA_{F(x_q)}\leqno(9.3)$$
for $x_0\in V(x)$, $x_1\in V(x_0),\ldots,x_q\in V(x_0,\ldots,x_{q-1})$.
We get in this way a morphism of complexes $F^\star:\cA^{[\bu]}(Y)\longrightarrow
(F^{-1}\cA)^{[\bu]}(X)$. On cohomology groups, we thus have an induced
morphism
$$F^\star~:~~H^q(Y,\cA)\longrightarrow H^q(X,F^{-1}\cA).\leqno(9.4)$$
Let $0\to\cA\to\cB\to\cC\to 0$ be an exact sequence of sheaves on $X$.
Thanks to property (9.1), there is an exact sequence
$$0\longrightarrow F^{-1}\cA\longrightarrow F^{-1}\cB\longrightarrow F^{-1}\cC\longrightarrow 0.$$
It is clear on the definitions that the morphism $F^\star$ in (9.4)
commutes with the associated cohomology exact sequences. Also, $F^\star$
preserves the cup product, i.e.\ $F^\star(\alpha\smallsmile\beta)=
F^\star\alpha\smallsmile F^\star\beta$ whenever $\alpha,\beta$ are
cohomology classes with values in sheaves $\cA$, $\cB$ on $X$.
Furthermore, if $G:Y\to Z$ is a continuous map, we have
$$(G\circ F)^\star=F^\star\circ G^\star.\leqno(9.5)$$

\claim{(9.6) Remark} \rm Similar definitions can be given for \v Cech
cohomology.
If $\cU=(U_\alpha)_{\alpha\in I}$ is an open covering of $Y$, then
$F^{-1}\cU=\big(F^{-1}(U_\alpha)\big)_{\alpha\in I}$ is an open covering of
$X$. For $c\in C^q(\cU,\cA)$, we set
$$(F^\star c)_{\alpha_0\ldots\alpha_q}=c_{\alpha_0\ldots\alpha_q}\circ F
\in C^q(F^{-1}\cU,F^{-1}\cA).$$
This definition is obviously compatible with the morphism from \v Cech
cohomology to ordinary cohomology.
\endclaim

\claim{(9.7) Remark} \rm Let $\Phi$ be a family of supports on $Y$. We define
$F^{-1}\Psi$ to be the family of closed sets $K\subset X$ such that $F(K)$
is contained in some set $L\in\Psi$. Then (9.4) can be generalized
in the form
$$F^\star~:~~H_\Psi^q(Y,\cA)\longrightarrow H_{F^{-1}\Psi}^q(X,F^{-1}\cA).
\leqno(9.8)$$
\endclaim

\claim{(9.9) Remark} \rm Assume that $X$ and $Y$ are paracompact differentiable
manifolds and that $F:X\to Y$ is a $\cC^\infty$ map. If $(\psi_\alpha)_{\alpha\in I}$
is a partition of unity subordinate to $\cU$, then $(\psi_\alpha\circ 
F)_{\alpha\in I}$ is a partition of unity on $X$ subordinate to $F^{-1}\cU$.
Let $c\in C^q(\cU,\bR)$. The differential form associated to $F^\star c$ in
the De Rham cohomology is
$$\eqalign{
g&=\sum_{\nu_0,\ldots,\nu_q}c_{\nu_0\ldots\nu_q}(\psi_{\nu_q}\circ F)
d(\psi_{\nu_0}\circ F)\wedge\ldots\wedge d(\psi_{\nu_{q-1}}\circ F)\cr
&=F^\star\Big(\sum_{\nu_0,\ldots,\nu_q}c_{\nu_0\ldots\nu_q}\,\psi_{\nu_q}
d\psi_{\nu_0}\wedge\ldots\wedge d\psi_{\nu_{q-1}}\Big).\cr}$$
Hence we have a commutative diagram
$$\cmalign{
&H^q_{\DR}(Y,\bR)&\buildo\simeq\over\longrightarrow&\check H^q(Y,\bR)
&\buildo\simeq\over\longrightarrow&H^q(Y,\bR)\cr
&\quad~~\big\downarrow F^\star&&\quad~~\big\downarrow F^\star
&&\quad~~\big\downarrow F^\star\cr
&H^q_{\DR}(X,\bR)&\buildo\simeq\over\longrightarrow&\check H^q(X,\bR)
&\buildo\simeq\over\longrightarrow&H^q(X,\bR).\cr}$$
\endclaim

\subsection{9.B}{Cohomology Groups of a Subspace}
Let $\cA$ be a sheaf on a topological space $X$, let $S$ be a subspace of $X$
and $i_S:S\lhra X$ the  inclusion. Then $i_S^{-1}\cA$ is the restriction of
$\cA$ to $S$, so that $H^q(S,\cA)=H^q(S,i_S^{-1}\cA)$ by definition. 
For any two subspaces $S'\subset S$, the inclusion of
$S'$ in $S$ induces a restriction morphism
$$H^q(S,\cA)\longrightarrow H^q(S',\cA).$$
    
\claim{(9.10) Theorem} Let $\cA$ be a sheaf on $X$ and $S$ a strongly
paracompact subspace in $X$. When $\Omega$ ranges over open neighborhoods of
$S$, we have 
$$H^q(S,\cA)=
\lim_{\displaystyle\longrightarrow\atop\scriptstyle\Omega\supset S}~~H^q(\Omega,\cA).$$
\endclaim

\proof{Proof} When $q=0$, the property is equivalent to Prop.\ 4.7.
The general case follows by induction on $q$ if we use the long cohomology 
exact sequences associated to the short exact sequence
$$0\longrightarrow\cA\longrightarrow\cA^{[0]}\longrightarrow\cA^{[0]}/\cA\longrightarrow 0$$
on $S$ and on its neighborhoods $\Omega$ (note that the restriction of
a flabby sheaf to $S$ is soft by Prop.\ 4.7 and the fact that
every closed subspace of a strongly paracompact subspace is strongly
paracompact).\qed
\endproof

\subsection{9.C}{Cartesian Product}
We use here the formalism of inverse images to deduce the cartesian 
product from the cup product. Let $R$ be a fixed commutative ring and
$\cA\to X$, $\cB\to Y$ sheaves of $R$-modules. We define the
{\it external tensor product} by
$$\cA\stimes_R\cB=\pr_1^{-1}\cA\otimes_R\pr_2^{-1}\cB
\leqno(9.11)$$
where $\pr_1$, $\pr_2$ are the projections of $X\times Y$ onto
$X$, $Y$ respectively. The sheaf $\cA\stimes_R\cB$ is thus the sheaf on 
$X\times Y$ whose stalks are
$$(\cA\stimes_R\cB)_{(x,y)}=\cA_x\otimes_R\cB_y.\leqno(9.12)$$
For all cohomology classes $\alpha\in H^p(X,\cA)$, $\beta\in H^q(Y,\cB)$
the {\it cartesian product} $\alpha\times\beta\in H^{p+q}(X\times Y,
\cA\stimes_R\cB)$ is defined by
$$\alpha\times\beta=(\pr_1^\star\alpha)\smallsmile(\pr_2^\star
\beta).\leqno(9.13)$$
More generally, let $\Phi$ and $\Psi$ be families of supports in $X$ and $Y$ 
respectively.
If $\Phi\times\Psi$ denotes the family of all closed subsets of 
$X\times Y$ contained in products $K\times L$ of elements $K\in\Phi$, 
$L\in\Psi$, the cartesian product defines a $R$-bilinear map
$$H_\Phi^p(X,\cA)\times H_\Psi^q(Y,\cB)\longrightarrow H_{\Phi\times\Psi}^{p+q}
(X\times Y,\cA\stimes_R\cB).\leqno(9.14)$$
If $\cA'\to X$, $\cB'\to Y$ are sheaves of abelian groups and if $\alpha'$,
$\beta'$ are cohomology classes of degree $p'$, $q'$ with values in $\cA'$, 
$\cB'$, one gets easily
$$(\alpha\times\beta)\smallsmile(\alpha'\times\beta')=(-1)^{qp'}
(\alpha\smallsmile\alpha')\times(\beta\smallsmile\beta').\leqno(9.15)$$
Furthermore, if $F:X'\to X$ and $G:Y'\to Y$ are continuous maps, then
$$(F\times G)^\star(\alpha\times\beta)=(F^\star\alpha)\times(G^\star\beta).
\leqno(9.16)$$

\section{10}{Spectral Sequence of a Filtered Complex}
\subsection{10.A}{Construction of the Spectral Sequence}
The theory of spectral sequences consists essentially in computing
the homo\-logy groups of a differential module $(K,d)$ by ``successive 
approximations", once a filtration $F_p(K)$ is given in $K$ 
and the cohomology groups of the graded modules $G_p(K)$ are known.
Let us first recall some standard definitions and notations concerning
filtrations.

\claim{(10.1) Definition}  Let $R$ be a commutative ring.  A filtration
of a $R$-module $M$ is a sequence of submodules $M_p\subset M$,
$p\in\bZ$, also denoted $M_p=F_p(M)$, such that $M_{p+1}\subset M_p$
for all $p\in\bZ$, $\bigcup M_p=M$ and $\bigcap M_p=\{0\}$.  The
associated graded module is
$$G(M)=\bigoplus_{p\in\bZ}G_p(M),~~~~G_p(M)=M_p/M_{p+1}.$$
\endclaim

Let $(K,d)$ be a differential module equipped with a filtration $(K_p)$
by differential submodules (i.e.\ $dK_p\subset K_p$ for every $p$).
For any number \hbox{$r\in\bN\cup\{\infty\}$}, we define
$Z^p_r,\,B^p_r\subset G_p(K)=K_p/K_{p+1}$ by
$$\leqalignno{
\qquad\qquad Z^p_r&=K_p\cap d^{-1}K_{p+r}~~\hbox{\rm mod}~K_{p+1},~~~
Z^p_\infty=K_p\cap d^{-1}\{0\}~~\hbox{\rm mod}~K_{p+1},&(10.2)\cr
\qquad\qquad B^p_r&=K_p\cap dK_{p-r+1}~~\hbox{\rm mod}~K_{p+1},~~~
B^p_\infty=K_p\cap dK~~~\hbox{\rm mod}~K_{p+1}.&(10.2')\cr}$$

\claim{(10.3) Lemma} For every $p$ and $r$, there are inclusions
$$\ldots\subset B^p_r\subset B^p_{r+1}\subset\ldots\subset B^p_\infty
\subset Z^p_\infty\subset\ldots\subset Z^p_{r+1}\subset Z^p_r\subset\ldots$$
and the differential $d$ induces an isomorphism
$$\wt d~:~~Z^p_r/Z^p_{r+1}\longrightarrow B^{p+r}_{r+1}/B^{p+r}_r.$$
\endclaim

\proof{Proof} It is clear that $(Z^p_r)$ decreases with $r$, that $(B^p_r)$ 
increases with $r$, and that $B^p_\infty\subset Z^p_\infty$. By definition
$$\eqalign{
Z^p_r&=(K_p\cap d^{-1}K_{p+r})/(K_{p+1}\cap d^{-1}K_{p+r}),\cr
B^p_r&=(K_p\cap dK_{p-r+1})/(K_{p+1}\cap dK_{p-r+1}).\cr}$$
The differential $d$ induces a morphism
$$Z^p_r\longrightarrow (dK_p\cap K_{p+r})/(dK_{p+1}\cap K_{p+r})$$
whose kernel is $(K_p\cap d^{-1}\{0\})/(K_{p+1}\cap d^{-1}\{0\})=Z^p_\infty$,
whence isomorphisms
$$\eqalign{
\wh d~:~~&Z^p_r/Z^p_\infty\longrightarrow(K_{p+r}\cap dK_p)/(K_{p+r}\cap dK_{p+1}),\cr
\wt d~:~~&Z^p_r/Z^p_{r+1}\longrightarrow(K_{p+r}\cap dK_p)/
(K_{p+r}\cap dK_{p+1}+K_{p+r+1}\cap dK_p).\cr}$$
The right hand side of the last arrow can be identified to 
$B^{p+r}_{r+1}/B^{p+r}_r$, for
$$\eqalignno{
B^{p+r}_r&=(K_{p+r}\cap dK_{p+1})/(K_{p+r+1}\cap dK_{p+1}),\cr
B^{p+r}_{r+1}&=(K_{p+r}\cap dK_p)/(K_{p+r+1}\cap dK_p).&\square\cr}$$
\endproof

Now, for each $r\in\bN$, we define a complex $E^\bu_r=\bigoplus_{p\in\bZ}
E^p_r$ with a differential $d_r:E^p_r\longrightarrow E^{p+r}_r$ of degree $r$
as follows: we set $E^p_r=Z^p_r/B^p_r$ and take
$$d_r~:~~Z^p_r/B^p_r\dlra Z^p_r/Z^p_{r+1}\buildo{\displaystyle\wt d}
\over\longrightarrow B^{p+r}_{r+1}/B^{p+r}_r\lhra Z^{p+r}_r/B^{p+r}_r\leqno(10.4)$$
where the first arrow is the obvious projection and the third arrow
the obvious inclusion. Since $d_r$ is induced by $d$, we actually
have $d_r\circ d_r=0$~; this can also be seen directly by the fact
that $B^{p+r}_{r+1}\subset Z^{p+r}_{r+1}$.

\claim{(10.5) Theorem and definition} There is a canonical 
isomorphism $E^\bu_{r+1}\simeq H^\bu(E^\bu_r)$. The sequence of differential
complexes $(E^\bu_r,d^\bu_r)$ is called the spectral sequence of
the filtered differential module $(K,d)$.
\endclaim

\proof{Proof} Since $\wt d$ is an isomorphism in (10.4), we have
$$\ker\,d_r=Z^p_{r+1}/B^p_r,~~~~\Im d_r=B^{p+r}_{r+1}/B^{p+r}_r.$$
Hence the image of $d_r:E^{p-r}_r\longrightarrow E^p_r$ is $B^p_{r+1}/B^p_r$ and
$$H^p(E^\bu_r)=(Z^p_{r+1}/B^p_r)/(B^p_{r+1}/B^p_r)
\simeq Z^p_{r+1}/B^p_{r+1}=E^p_{r+1}.\eqno\square$$
\endproof

\claim{(10.6) Theorem} Consider the filtration of the homology module
$H(K)$ defined by
$$F_p\big(H(K)\big)=\Im\big(H(K_p)\longrightarrow H(K)\big).$$
Then there is a canonical isomorphism
$$E^p_\infty=G_p\big(H(K)\big).$$
\endclaim

\proof{Proof} Clearly $F_p\big(H(K)\big)=(K_p\cap d^{-1}\{0\})/(K_p\cap dK)$,
whereas
$$\eqalign{
Z^p_\infty&=(K_p\cap d^{-1}\{0\})/(K_{p+1}\cap d^{-1}\{0\}),~~
B^p_\infty=(K_p\cap dK)/(K_{p+1}\cap dK),\cr
E^p_\infty&=Z^p_\infty/B^p_\infty=
(K_p\cap d^{-1}\{0\})/(K_{p+1}\cap d^{-1}\{0\}+K_p\cap dK).\cr}$$
It follows that $E^p_\infty\simeq F_p\big(H(K)\big)/
F_{p+1}\big(H(K)\big)$.\qed
\endproof

In most applications, the differential module $K$ has a natural grading
compatible with the filtration. Let us consider for example the case
of a cohomology complex $K^\bu=\bigoplus_{l\in\bZ}K^l$. The filtration
$K^\bu_p=F_p(K^\bu)$ is said to be {\it compatible} with the
differential complex structure if each $K^\bu_p$ is a subcomplex of
$K^\bu$, i.e.\
$$K^\bu_p=\bigoplus_{l\in\bZ}K^l_p$$
where $(K^l_p)$ is a filtration of $K^l$. Then we define
$Z^{p,q}_r$, $B^{p,q}_r$, $E^{p,q}_r$ to be the sets of elements of
$Z^p_r$, $B^p_r$, $E^p_r$ of total degree $p+q$. Therefore
\medskip\noindent
$\cmalign{
\noalign{\vskip2pt}
(10.7)\kern40pt\hfill&Z^{p,q}_r&=K^{p+q}_p\cap d^{-1}K^{p+q+1}_{p+r}
~~~\hbox{\rm mod}~~K^{p+q}_{p+1}~,~~~~&Z^p_r=\bigoplus Z^{p,q}_r,\cr
\noalign{\vskip4pt}
(10.7')\hfill&B^{p,q}_r&=K^{p+q}_p\cap dK^{p+q-1}_{p-r+1}
~~~\hbox{\rm mod}~~K^{p+q}_{p+1}~,~~~~&B^p_r=\bigoplus B^{p,q}_r,\cr
\noalign{\vskip5pt}
(10.7'')\hfill&E^{p,q}_r&=Z^{p,q}_r/B^{p,q}_r~,&E^p_r=\bigoplus E^{p,q}_r,\cr}$
\medskip\noindent
and the differential $d_r$ has bidegree $(r,-r+1)$, i.e.\
$$d_r~:~~E^{p,q}_r\longrightarrow E^{p+r\,,\,q-r+1}_r.\leqno(10.8)$$
For an element of pure bidegree $(p,q)$, $p$ is called the
{\it filtering degree}, $q$ the {\it complementary degree} and $p+q$ the
{\it total degree}.

\claim{(10.9) Definition} A filtration $(K^\bu_p)$ of a complex $K^\bu$
is said to be regular if for each degree $l$ there are indices
$\nu(l)\le N(l)$ such that $K^l_p=K^l$ for $p<\nu(l)$ and
$K^l_p=0$ for $p>N(l)$.
\endclaim

If the filtration is regular, then (10.7) and $(10.7')$ show that
$$\eqalign{
Z^{p,q}_r=Z^{p,q}_{r+1}=\ldots=Z^{p,q}_\infty~~~~\hbox{\rm for}~~
&r>N(p+q+1)-p,\cr
B^{p,q}_r=B^{p,q}_{r+1}=\ldots=B^{p,q}_\infty~~~~\hbox{\rm for}~~
&r>p+1-\nu(p+q-1),\cr}$$
therefore $E^{p,q}_r=E^{p,q}_\infty$ for $r\ge r_0(p,q)$. We say that
the spectral sequence {\it converges} to its limit term, and we write
symbolically
$$E^{p,q}_r\Longrightarrow H^{p+q}(K^\bu)\leqno(10.10)$$
to express the following facts: there is a spectral sequence whose terms of
the $r$-th generation are $E^{p,q}_r$, the sequence converges to a limit term
$E^{p,q}_\infty$, and $E^{p,l-p}_\infty$ is the term $G_p\big(H^l(K^\bu)\big)$ 
in the graded module associated to some filtration of $H^l(K^\bu)$.

\claim{(10.11) Definition} The spectral sequence is said to 
collapse in $E^\bu_r$ if all terms $Z^{p,q}_k$, $B^{p,q}_k$,
$E^{p,q}_k$ are constant for $k\ge r$, or equivalently if $d_k=0$ 
in all bidegrees for $k\ge r$.
\endclaim

\claim{(10.12) Special case} \rm Assume that there exists an integer
$r\ge 2$ and an index $q_0$ such that $E^{p,q}_r=0$ for $q\ne q_0$. Then this
property remains true for larger values of $r$, and we must have $d_r=0$. 
It follows that the spectral sequence collapses in $E^\bu_r$ and that 
$$H^l(K^\bu)=E^{l-q_0,q_0}_r.$$
Similarly, if $E^{p,q}_r=0$ for $p\ne p_0$ and some $r\ge 1$ then 
$$H^l(K^\bu)=E^{p_0,l-p_0}_r.\eqno\square$$
\endclaim

\subsection{10.B}{Computation of the First Terms} 
Consider an arbitrary spectral sequence.
For  $r=0$, we have $Z^p_0=K_p/K_{p+1}$, $B^p_0=\{0\}$, thus 
$$E^p_0=K_p/K_{p+1}=G_p(K).\leqno(10.13)$$
The differential $d_0$ is induced by $d$ on the quotients, and 
$$E^p_1=H\big(G_p(K)\big).\leqno(10.14)$$
Now, there is a short exact sequence of differential modules
$$0\longrightarrow G_{p+1}(K)\longrightarrow K_p/K_{p+2}\longrightarrow G_p(K)\longrightarrow 0.$$
We get therefore a connecting homomorphism
$$E^p_1=H\big(G_p(K)\big)\buildo\partial\over\longrightarrow
H\big(G_{p+1}(K)\big)=E^{p+1}_1.\leqno(10.15)$$
We claim that $\partial$ coincides with the differential $d_1$~: indeed,
both are induced by $d$. When $K^\bu$ is a filtered cohomology complex,
$d_1$ is the connecting homomorphism
$$E^{p,q}_1=H^{p+q}\big(G_p(K^\bu)\big)\buildo\partial\over\longrightarrow
H^{p+q+1}\big(G_{p+1}(K^\bu)\big)=E^{p+1,q}_1.\leqno(10.16)$$

\section{11}{Spectral Sequence of a Double Complex}
A double complex is a bigraded module $K^{\bu,\bu}=\bigoplus K^{p,q}$
together with a differential $d=d'+d''$ such that
$$d':K^{p,q}\longrightarrow K^{p+1,q},~~~d'':K^{p,q+1}\longrightarrow K^{p,q+1},\leqno(11.1)$$
and $d\circ d=0$. In particular, $d'$ and $d''$ satisfy the relations
$$d^{\prime2}=d^{\prime\prime2}=0,~~~d'd''+d''d'=0.\leqno(11.2)$$
The {\it simple complex associated} to $K^{\bu,\bu}$ is defined by 
$$K^l=\bigoplus_{p+q=l}K^{p,q}$$
together with the differential $d$. We will suppose here
that both graduations of $K^{\bu,\bu}$ are positive, i.e.\ $K^{p,q}=0$
for $p<0$ or $q<0$. The {\it first filtration} of $K^\bu$ is
defined by
$$K^l_p=\bigoplus_{i+j=l,~i\ge p}K^{i,j}=\bigoplus_{p\le i\le l}K^{i,l-i}.
\leqno(11.3)$$
The graded module associated to this filtration is of course 
$G_p(K^l)=K^{p,l-p}$, and the differential induced by $d$ on the quotient 
coincides with $d''$ because $d'$ takes $K^l_p$ to $K^{l+1}_{p+1}$. 
Thus we have a spectral sequence beginning by
$$E^{p,q}_0=K^{p,q},~~~d_0=d'',~~~E^{p,q}_1=H^q_{d''}(K^{p,\bu}).
\leqno(11.4)$$
By (10.16), $d_1$ is the connecting homomorphism associated to the
short exact sequence
$$0\longrightarrow K^{p+1,\bu}\longrightarrow K^{p,\bu}\oplus K^{p+1,\bu}\longrightarrow K^{p,\bu}\longrightarrow 0$$
where the differential is given by ($d$ mod $K^{p+2,\bu}$)
for the central term and by $d''$ for the two others. The definition of the 
connecting homomorphism in the proof of Th. 1.5 shows that 
$$d_1=\partial:~~H^q_{d''}(K^{p,\bu})\longrightarrow H^q_{d''}(K^{p+1,\bu})$$
is induced by $d'$. Consequently, we find
$$E^{p,q}_2=H^p_{d'}(E_1^{\bu,q})=
H^p_{d'}\big(H^q_{d''}(K^{\bu,\bu})\big).\leqno(11.5)$$

For such a spectral sequence, several interesting additional
features can be pointed out. For all $r$ and $l$, there is an injective 
homomorphism
$$E^{0,l}_{r+1}\lhra E^{0,l}_r$$
whose image can be identified with the set of $d_r$-cocycles in
$E^{0,l}_r$~; the coboundary group is zero because $E^{p,q}_r=0$ for
$q<0$. Similarly, $E^{l,0}_r$ is equal to its cocycle submodule, and there
is a surjective homomorphism
$$E^{l,0}_r\dlra E^{l,0}_{r+1}\simeq E^{l,0}_r/d_rE^{l-r,r-1}_r.$$
Furthermore, the filtration on $H^l(K^\bu)$ begins at $p=0$ and stops  
at $p=l$, i.e.\ 
$$F_0\big(H^l(K^\bu)\big)=H^l(K^\bu),~~~F_p\big(H^l(K^\bu)\big)=0~~~
\hbox{\rm for}~~p>l.\leqno(11.6)$$
Therefore, there are canonical maps
$$\eqalign{
&H^l(K^\bu)\dlra G_0\big(H^l(K^\bu)\big)=E^{0,l}_\infty\lhra E^{0,l}_r,\cr
&E^{l,0}_r\dlra E^{l,0}_\infty=G_l\big(H^l(K^\bu)\big)\lhra H^l(K^\bu).}
\leqno(11.7)$$
These maps are called the {\it edge homomorphisms} of the spectral sequence.

\claim{(11.8) Theorem} There is an exact sequence
$$0\longrightarrow E^{1,0}_2\longrightarrow H^1(K^\bu)\longrightarrow E^{0,1}_2\buildo d_2\over\longrightarrow E^{2,0}_2
\longrightarrow H^2(K^\bu)$$
where the non indicated arrows are edge homomorphisms.
\endclaim

\proof{Proof} By 11.6, the graded module associated to $H^1(K^\bu)$ has only
two components, and we have an exact sequence
$$0\longrightarrow E^{1,0}_\infty\longrightarrow H^1(K^\bu)\longrightarrow E^{0,1}_\infty\longrightarrow 0.$$
However $E^{1,0}_\infty=E^{1,0}_2$ because all differentials $d_r$
starting from $E^{1,0}_r$ or abuting to $E^{1,0}_r$ must be zero
for $r\ge 2$. Similarly, $E^{0,1}_\infty=E^{0,1}_3$ and 
$E^{2,0}_\infty=E^{2,0}_3$, thus there is an exact sequence
$$0\longrightarrow E^{0,1}_\infty\longrightarrow E^{0,1}_2\buildo d_2\over\longrightarrow E^{2,0}_2\longrightarrow
E^{2,0}_\infty\longrightarrow 0.$$
A combination of the two above exact sequences yields
$$0\longrightarrow E^{1,0}_2\longrightarrow H^1(K^\bu)\longrightarrow E^{0,1}_2\buildo d_2\over\longrightarrow E^{2,0}_2\longrightarrow
E^{2,0}_\infty\longrightarrow 0.$$
Taking into account the injection $E^{2,0}_\infty\lhra H^2(K^\bu)$ in 
(11.7), we get the required exact sequence.\qed
\endproof

\claim{(11.9) Example} \rm Let $X$ be a complex manifold of dimension $n$.
Consider the double complex $K^{p,q}=\cC^\infty_{p,q}(X,\bC)$ together with the
exterior derivative $d=d'+d''$. Then there is a spectral sequence
which starts from the Dolbeault cohomology groups
$$E^{p,q}_1=H^{p,q}(X,\bC)$$ 
and which converges to the graded module associated to a filtration of 
the De Rham cohomology groups:
$$E^{p,q}_r\Longrightarrow H^{p+q}_{\DR}(X,\bC).$$
This spectral sequence is called the {\it Hodge-Fr\"olicher spectral 
sequence} \cite{Fr\"o\-licher 1955}. We will study it in much more
detail in chapter 6 when $X$ is compact.\qed
\endclaim

Frequently, the spectral sequence under consideration can be obtained from
two distinct double complexes and one needs to compare the final
cohomology groups. The following lemma can often be applied.

\claim{(11.10) Lemma} Let $K^{p,q}\longrightarrow L^{p,q}$ be a morphism of
double complexes $($i.e.\ a double sequence of maps commuting with $d'$ and 
$d'')$. Then there are induced morphisms 
$${}_KE^{\bu,\bu}_r\longrightarrow{}_LE^{\bu,\bu}_r,~~~~r\ge 0$$
of the associated spectral sequences. If one of these morphisms is an 
isomorphism for some $r$, then $H^l(K^\bu)\longrightarrow H^l(L^\bu)$
is an isomorphism.
\endclaim

\proof{Proof} If the $r$-terms are isomorphic, they have the same
cohomology groups, thus ${}_KE^{\bu,\bu}_{r+1}\simeq{}_LE^{\bu,\bu}_{r+1}$
and ${}_KE^{\bu,\bu}_\infty\simeq{}_LE^{\bu,\bu}_\infty$ in the limit.
The lemma follows from the fact that if a morphism of graded modules
$\varphi:M\longrightarrow M'$ induces an isomorphism
$G_\bu(M)\longrightarrow G_\bu(M')$, then $\varphi$ is an isomorphism.\qed
\endproof

\section{12}{Hypercohomology Groups}
Let $(\cL^\bu,\delta)$ be a complex of sheaves
$$0\longrightarrow\cL^0\buildo\delta^0\over\longrightarrow\cL^1\longrightarrow\cdots\longrightarrow\cL^q\buildo\delta^q
\over\longrightarrow\cdots$$
on a topological space $X$. We denote by $\cH^q=\cH^q(\cL^\bu)$ the
$q$-th sheaf of cohomology of this complex; thus $\cH^q$ is a sheaf of
abelian groups over $X$. Our goal is to define ``generalized
cohomology groups'' attached to $\cL^\bu$ on~$X$, in such a way
that these groups only depend on the cohomology sheaves~$\cH^q$.
For this, we associate to $\cL^\bu$ the double complex of groups
$$K_\cL^{p,q}=(\cL^q)^{[p]}(X)\leqno(12.1)$$
with differential $d'=d^p$ given by (2.5), and with $d''=(-1)^p
(\delta^q)^{[p]}$. As~$(\delta^q)^{[\bu]}:(\cL^q)^{[\bu]}\longrightarrow
(\cL^{q+1})^{[\bu]}$ is a morphism of complexes, we get the expected
relation $d'd''+d''d'=0$. 

\claim{(12.2) Definition} The groups $H^q(K_\cL^\bu)$ are called the
hypercohomology groups of $\cL^\bu$ and are denoted $\bH^q(X,\cL^\bu)$.
\endclaim

Clearly $\bH^0(X,\cL^\bu)=\cH^0(X)$ where $\cH^0=\ker\,\delta^0$ is
the first cohomology sheaf of $\cL^\bu$.
If $\varphi^\bu:\cL^\bu\longrightarrow\cM^\bu$ is a morphism of sheaf complexes, there is
an associated morphism of double complexes
$\varphi^{\bu,\bu}:K^{\bu,\bu}_\cL\longrightarrow K^{\bu,\bu}_\cM$, hence a natural
morphism
$$\bH^q(\varphi^\bu)~:~~\bH^q(X,\cL^\bu)\longrightarrow\bH^q(X,\cM^\bu).
\leqno(12.3)$$
We first list a few immediate properties of hypercohomology groups,
whose proofs are left to the reader.

\claim{(12.4) Proposition} The following properties hold:
\medskip
\item{\rm a)} If $\cL^q=0$ for $q\ne 0$,
then $\bH^q(X,\cL^\bu)=H^q(X,\cL^0)$.
\medskip
\item{\rm b)} If $\cL^\bu[s]$ denotes the complex $\cL^\bu$ shifted
of $s$ indices to the right, i.e.\ $\cL^\bu[s]^q=\cL^{q-s}$, then
$\bH^q(X,\cL^\bu[s])=\bH^{q-s}(X,\cL^\bu)$.
\medskip
\item{\rm c)} If $0\longrightarrow\cL^\bu\longrightarrow\cM^\bu\longrightarrow\cN^\bu\longrightarrow0$ is an exact
sequence of sheaf complexes, there is a long exact sequence
$$\cdots\bH^q(X,\cL^\bu)\longrightarrow\bH^q(X,\cM^\bu)\longrightarrow\bH^q(X,\cN^\bu)
\buildo\partial\over\longrightarrow\bH^{q+1}(X,\cL^\bu)\cdots.\eqno\square$$
\smallskip
\endclaim

If $\cL^\bu$ is a sheaf complex, the spectral sequence associated to the
first filtration of $K_\cL^\bu$ is given by
$$E^{p,q}_1=H^q_{d''}(K_\cL^{p,\bu})=H^q\big((\cL^\bu)^{[p]}(X)\big).$$
However by (2.9) the functor $\cA\longmapsto\cA^{[p]}(X)$ preserves
exact sequences. Therefore, we get
$$\leqalignno{
E^{p,q}_1&=\big(\cH^q(\cL^\bu)\big)^{[p]}(X),&(12.5)\cr
E^{p,q}_2&=H^p\big(X,\cH^q(\cL^\bu)\big),&(12.5')\cr}$$
since $E_2^{p,q}=H^p_{d'}(E_1^{\bu,q})$. If $\varphi^\bu:\cL^\bu\longrightarrow\cM^\bu$
is a morphism, an application of Lemma 11.10 to the $E_2$-term of the
associated first spectral sequences of $K_\cL^{\bu,\bu}$ and
$K_\cM^{\bu,\bu}$ yields:

\claim{(12.6) Corollary} If $\varphi^\bu:\cL^\bu\longrightarrow\cM^\bu$ is a
quasi-isomorphism $\big($this means that $\varphi^\bu$ induces an
isomorphism $\cH^\bu(\cL^\bu)\longrightarrow\cH^\bu(\cM^\bu)~\big)$, then
$$\bH^l(\varphi^\bu)~:~~\bH^l(X,\cL^\bu)\longrightarrow\bH^l(X,\cM^\bu)$$
is an isomorphism.
\endclaim

Now, we may reverse the roles of the indices $p,q$ and of the
differentials $d',d''$. The {\it second filtration} 
$F_p(K^l_\cL)=\bigoplus_{j\ge p}K_\cL^{l-j,j}$ is associated to a spectral
sequence such that $\smash{\wt E}^{p,q}_1=H^q_{d'}(K_\cL^{\bu,p})=
\smash{H^q_{d'}\big((\cL^p)^{[\bu]}(X)\big)}$, hence
$$\leqalignno{
\wt E^{p,q}_1&=H^q(X,\cL^p),&(12.7)\cr
\wt E^{p,q}_2&=H^p_{\delta}\big(H^q(X,\cL^\bu)\big).&(12.7')\cr}$$
These two spectral sequences converge to limit terms which are the graded
modules associated to filtrations of $\bH^\bu(X,\cL^\bu)$~; these
filtrations are in gene\-ral different.
Let us mention two interesting special cases.
\medskip
\noindent{$\bullet$} Assume first that the complex $\cL^\bu$ is a resolution
of a sheaf $\cA$, so that $\cH^0=\cA$ and $\cH^q=0$ for $q\ge 1$. 
Then we find 
$$E^{p,0}_2=H^p(X,\cA),~~~~E^{p,q}_2=0~~\hbox{\rm for}~~q\ge 1.$$
It follows that the first spectral sequence collapses in $E^\bu_2$, and
10.12 implies
$$\bH^l(X,\cL^\bu)\simeq H^l(X,\cA).\leqno(12.8)$$

\noindent{$\bullet$} Now, assume that the sheaves $\cL^q$ are acyclic.
The second spectral sequence gives
$$\leqalignno{
\wt E^{p,0}_2=H^p&\big(\cL^\bu(X)\big),~~~~
\wt E^{p,q}_2=0~~\hbox{\rm for}~~q\ge 1,\cr
&\bH^l(X,\cL^\bu)\simeq H^l\big(\cL^\bu(X)\big).&(12.9)\cr}$$

If both conditions hold, i.e.\ if $\cL^\bu$ is a resolution of a sheaf $\cA$
by acyclic sheaves, then (12.8) and (12.9) can be combined to obtain
a new proof of the De Rham-Weil isomorphism
$H^l(X,\cA)\simeq H^l\big(\cL^\bu(X)\big)$.

\section{13}{Direct Images and the Leray Spectral Sequence}
\subsection{13.A}{Direct Images of a Sheaf}
Let $X,Y$ be topological spaces, $F:X\to Y$ a continuous map and $\cA$ a sheaf
of abelian groups on $X$. Recall that the {\it direct image} $F_\star\cA$ is
the presheaf on $Y$ defined for any open set $U\subset Y$ by
$$(F_\star\cA)(U)=\cA\big(F^{-1}(U)\big).\leqno(13.1)$$
Axioms (II-$2.4'$ and (II-$2.4'')$ are clearly satisfied, thus $F_\star\cA$
is in fact a sheaf. The following result is obvious:
$$\cA~~\hbox{\rm is flabby}~~~\Longrightarrow~~~F_\star\cA~~
\hbox{\rm is flabby.}\leqno(13.2)$$
Every sheaf morphism $\varphi:\cA\to\cB$ induces a corresponding morphism 
$$F_\star\varphi~:~~F_\star\cA\longrightarrow F_\star\cB,$$
so $F_\star$ is a functor on the category of sheaves on $X$ to the category of
sheaves on $Y$. This functor is exact on the left: indeed, to every exact 
sequence $0\to\cA\to\cB\to\cC$ is associated an exact sequence
$$0\longrightarrow F_\star\cA\longrightarrow F_\star\cB\longrightarrow F_\star\cC,$$
but $F_\star\cB\to F_\star\cC$ need not be onto if $\cB\to\cC$ is. All this
follows immediately from the considerations of \S 3. In particular, the 
simplicial flabby resolution $(\cA^{[\bu]},d)$ yields a complex of sheaves
$$0\longrightarrow F_\star\cA^{[0]}\longrightarrow F_\star\cA^{[1]}\longrightarrow\cdots\longrightarrow
F_\star\cA^{[q]}\buildo F_\star d^q\over\longrightarrow F_\star\cA^{[q+1]}\longrightarrow\cdots.
\leqno(13.3)$$

\claim{(13.4) Definition} The $q$-th direct image of $\cA$ by $F$ is the
$q$-th cohomology sheaf of the sheaf complex $(13.3)$. It is denoted
$$R^qF_\star\cA=\cH^q(F_\star\cA^{[\bu]}).$$
\endclaim

As $F_\star$ is exact on the left, the sequence $0\to F_\star\cA\to
F_\star\cA^{[0]}\to F_\star\cA^{[1]}$ is exact, thus
$$R^0F_\star\cA=F_\star\cA.\leqno(13.5)$$
We now compute the stalks of $R^qF_\star\cA$. As the kernel or cokernel
of a sheaf morphism is obtained stalk by stalk, we have
$$(R^qF_\star\cA)_y=H^q\big((F_\star\cA^{[\bu]})_y\big)=
\lim_{\displaystyle\longrightarrow\atop\scriptstyle U\ni y}~~
H^q\big(F_\star\cA^{[\bu]}(U)\big).$$
The very definition of $F_\star$ and of sheaf cohomology groups implies
$$H^q\big(F_\star\cA^{[\bu]}(U)\big)=H^q\big(\cA^{[\bu]}(F^{-1}(U))\big)=
H^q\big(F^{-1}(U),\cA\big),$$
hence we find
$$(R^qF_\star\cA)_y=\lim_{\displaystyle\longrightarrow\atop\scriptstyle U\ni y}~~
H^q\big(F^{-1}(U),\cA\big),\leqno(13.6)$$
i.e.\ $R^qF_\star\cA$ is the sheaf associated to the presheaf
$U\mapsto H^q\big(F^{-1}(U),\cA\big)$. We~must stress here that the stronger 
relation
$R^qF_\star\cA(U)=H^q\big(F^{-1}(U),\cA\big)$ need not be true in general.
If the fiber $F^{-1}(y)$ is strongly paracompact in $X$ and if the
family of open sets $F^{-1}(U)$ is a fundamental family of
neighborhoods of $F^{-1}(y)$ (this situation occurs for example if
$X$ and $Y$ are locally compact spaces and $F$ a proper map, or if
$F=\pr_1:X=Y\times S\longrightarrow Y$ where $S$ is compact), Th. 9.10
implies the more natural relation
$$(R^qF_\star\cA)_y=H^q\big(F^{-1}(y),\cA\big).\leqno(13.6')$$

Let $0\to\cA\to\cB\to\cC\to 0$ be an exact sequence of sheaves on $X$.
Apply the long exact sequence of cohomology on every open set $F^{-1}(U)$
and take the direct limit over~$U$. We get an exact sequence of sheaves:
$$\cmalign{
\noalign{\vskip2pt}
&~~~0~&~\longrightarrow~~~F_\star\cA&~\longrightarrow~~~F_\star\cB&~\longrightarrow~~~F_\star\cC&~\longrightarrow~~~
R^1F_\star\cA&~\longrightarrow~\cdots\cr
\noalign{\vskip2pt}
&\cdots&~\longrightarrow~ R^qF_\star\cA&~\longrightarrow~ R^q F_\star\cB&~\longrightarrow~ R^q F_\star\cC&~\longrightarrow~
R^{q+1}F_\star\cA&~\longrightarrow~\cdots.\cr}
\leqno{\displaystyle(13.7)\atop\strut}$$

\subsection{13.B}{Leray Spectral Sequence}
For any continuous map~$F:X\to Y$, the Leray spectral sequence relates
the cohomology groups of a sheaf $\cA$ on $X$ and those of its direct
images $R^qF_\star\cA$ on~$Y$. Consider the two spectral
sequences $E^\bu_r$, $\wt E^\bu_r$ associated with the complex of sheaves
$\cL^\bu=F_\star\cA^{[\bu]}$ on~$Y$, as in \S~12. By definition we have
$\cH^q(\cL^\bu)=R^qF_\star\cA$. By $(12.5')$ the second term of the first
spectral sequence is
$$E^{p,q}_2=H^p(Y,R^qF_\star\cA),$$
and this spectral sequence converges to the graded module associated
to a filtration of $\bH^l(Y,F_\star\cA^{[\bu]})$.
On the other hand, (13.2) implies that $F_\star\cA^{[q]}$ is flabby. 
Hence, the second special case (12.9) can be applied:
$$\bH^l(Y,F_\star\cA^{[\bu]})\simeq H^l\big(F_\star\cA^{[\bu]}(Y)\big)
=H^l\big(\cA^{[\bu]}(X)\big)=H^l(X,\cA).$$
We may conclude this discussion by the following

\claim{(13.8) Theorem} For any continuous map $F:X\to Y$ and any
sheaf $\cA$ of abelian groups on $X$, there exists a spectral sequence
whose $E^\bu_2$ term is
$$E_2^{p,q}=H^p(Y,R^qF_\star\cA),$$
which converges to a limit term $E^{p,l-p}_\infty$ equal to the graded 
module associated with a filtration of $H^l(X,\cA)$. The edge homomorphism
$$H^l(Y,F_\star\cA)\dlra E^{l,0}_\infty\lhra H^l(X,\cA)$$
coincides with the composite morphism
$$F^{\#}:~~H^l(Y,F_\star\cA)\buildo F^\star\over\longrightarrow H^l(X,F^{-1}F_\star\cA)
\buildo H^l(\mu_F)\over\llra H^l(X,\cA)$$
where $\mu_F:F^{-1}F_\star\cA\longrightarrow\cA$ is the canonical sheaf morphism.
\endclaim

\proof{Proof} Only the last statement remains to be proved. The morphism
$\mu_F$ is defined as follows: every element $s\in (F^{-1}F_\star\cA)_x=
(F_\star\cA)_{F(x)}$ is the germ of a section 
$\wt s\in F_\star\cA(V)=\cA\big(F^{-1}(V)\big)$ on a neighborhood
$V$ of $F(x)$. Then $F^{-1}(V)$ is a neighborhood of $x$ and we let
$\mu_F s$ be the germ of $\wt s$ at $x$.

Now, we observe that to any commutative diagram of topological spaces
and continuous maps is associated a commutative diagram involving the
direct image sheaves and their cohomology groups:
$$\cmalign{
\hfill X&\buildo F\over\longrightarrow&~Y\kern80pt\hfill
H^l(X,\cA)&\buildo~~F^\#\over\longleftarrow &H^l(Y,F_\star\cA)\cr
{\scriptstyle G}\big\downarrow&&~\big\downarrow{\scriptstyle H}\hfill
{\scriptstyle G^\#}\big\uparrow\qquad&&\qquad\big\uparrow
{\scriptstyle H^\#}\cr
\hfill X'&\buildo F'\over\longrightarrow&~Y'\hfill
H^l(X',G_\star\cA)&\buildo~~F^{\prime\#}\over\longleftarrow 
&H^l(Y',F'_\star G_\star\cA).\cr}$$
There is a similar commutative diagram in which $F^\#$ and $F^{\prime\#}$ are
replaced by the edge homomorphisms of the spectral sequences of
$F$ and $F'$~: indeed there is a natural morphism
$H^{-1}F'_\star\cB\longrightarrow F_\star G^{-1}\cB$ for any sheaf $\cB$ on $X'$,
so we get a morphism of sheaf complexes
$$H^{-1}F'_\star(G_\star\cA)^{[\bu]}\longrightarrow
F_\star G^{-1}(G_\star\cA)^{[\bu]}\longrightarrow
F_\star(G^{-1}G_\star\cA)^{[\bu]}\longrightarrow F_\star\cA^{[\bu]},$$
hence also a morphism of the spectral sequences associated to both ends.

The special case $X'=Y'=Y$, $G=F$, $F'=H=\Id_Y$ then shows
that our statement is true for $F$ if it is true for $F'$. Hence
we may assume that $F$ is the identity map; in this case, we need
only show that the edge homomorphism of the spectral sequence of
$F_\star\cA^{[\bu]}=\cA^{[\bu]}$ is the identity map. This is an 
immediate consequence of the fact that we have a quasi-isomorphism 
$$(\cdots\to 0\to\cA\to 0\to\cdots)\longrightarrow\cA^{[\bu]}.\eqno\square$$
\endproof

\claim{(13.9) Corollary} If $R^qF_\star\cA=0$ for $q\ge 1$, there is
an isomorphism $H^l(Y,F_\star\cA)\simeq H^l(X,\cA)$ induced
by $F^{\#}$.
\endclaim

\proof{Proof} We are in the special case 10.12 with $E^{p,q}_2=0$ for $q\ne 0$, so
$$H^l(Y,F_\star\cA)=E^{l,0}_2\simeq H^l(X,\cA).\eqno\square$$
\endproof

\claim{(13.10) Corollary} Let $F:X\longrightarrow Y$ be a proper finite-to-one map.
For any sheaf $\cA$ on $X$, we have $R^qF_\star\cA=0$ for $q\ge 1$ 
and there is an isomorphism $H^l(Y,F_\star\cA)\simeq H^l(X,\cA)$.
\endclaim

\proof{Proof} By definition of higher direct images, we have
$$(R^qF_\star\cA)_y=\lim_{\displaystyle\,\longrightarrow\atop\scriptstyle{U\ni y}}
~~H^q\big(\cA^{[\bu]}\big(F^{-1}(U)\big)\big).$$
If $F^{-1}(y)=\{x_1,\ldots,x_m\}$, the assumptions imply that
$\big(F^{-1}(U)\big)$ is a fundamental system of neighborhoods of
$\{x_1,\ldots,x_m\}$. Therefore
$$(R^qF_\star\cA)_y=\bigoplus_{1\le j\le m}H^q\big(\cA^{[\bu]}_{x_j}\big)
=\cases{\bigoplus\cA_{x_j}&for~~$q=0$,\cr 0&for~~$q\ge 1$,\cr}$$
and we conclude by Cor.\ 13.9.\qed
\endproof

Corollary 13.10 can be applied in particular to the inclusion
$j:S\to X$ of a {\it closed} subspace $S$. Then $j_\star\cA$ coincides
with the sheaf $\cA^S$ defined in \S 3 and we get
$H^q(S,\cA)=H^q(X,\cA^S)$. It is very important to observe that the
property $R^qj_\star\cA=0$ for $q\ge 1$ need not be true if $S$
is not closed.

\subsection{13.C}{Topological Dimension}
As a first application of the Leray spectral sequence, we are going to 
derive some properties related to the concept of {\it topological dimension}.

\claim{(13.11) Definition} A non empty space $X$ is said to be of topological
dimension $\le n$ if $H^q(X,\cA)=0$ for any $q>n$ and any sheaf $\cA$ on $X$.
We let $\topdim X$ be the smallest such integer $n$ if it exists, and
$+\infty$ otherwise.
\endclaim
  
\claim{(13.12) Criterion} For a paracompact space $X$, the following 
conditions are equivalent:
\medskip
\item{\rm a)} $\topdim X\le n~;$
\medskip
\item{\rm b)} the sheaf $\cZ^n=\ker(\cA^{[n]}\longrightarrow\cA^{[n+1]})$
is soft for every sheaf $\cA~;$
\medskip
\item{\rm c)} every sheaf $\cA$ admits a resolution $0\to\cL^0\to\cdots
\to\cL^n\to 0$ of length $n$ by soft sheaves.\smallskip
\endclaim

\proof{Proof} b) $\Longrightarrow$ c). Take $\cL^q=\cA^{[q]}$ for $q<n$ and
$\cL^n=\cZ^n$.
\medskip
\noindent{c)} $\Longrightarrow$ a). For every sheaf $\cA$, 
the De Rham-Weil isomorphism implies $H^q(X,\cA)=H^q\big(\cL^\bu(X)\big)=0$
when $q>n$.
\medskip
\noindent{a)} $\Longrightarrow$ b). Let $S$ be a closed set and 
$U=X\ssm S$. As in Prop.\ 7.12,
$(\cA^{[\bu]})_U$ is an acyclic resolution of $\cA_U$. Clearly
$\ker\big((\cA^{[n]})_U\to(\cA^{[n+1]})_U\big)=\cZ^n_U$, so the isomorphisms
(6.2) obtained in the proof of the De Rham-Weil theorem imply
$$H^1(X,\cZ^n_U)\simeq H^{n+1}(X,\cA_U)=0.$$
By (3.10), the restriction map $\cZ^n(X)\longrightarrow\cZ^n(S)$ is onto, so
$\cZ^n$ is soft.\qed
\endproof

\claim{(13.13) Theorem} The following properties hold:
\medskip
\item{\rm a)} If $X$ is paracompact and if every point of $X$ has 
a neighborhood of topological dimension $\le n$, then $\topdim X\le n$.
\medskip
\item{\rm b)} If $S\subset X$, then $\topdim S\le \topdim X$
provided that $S$ is closed or $X$ metrizable.
\medskip
\item{\rm c)}If $X,Y$ are metrizable spaces, one of them locally 
compact, then 
$$\topdim(X\times Y)\le\topdim X+\topdim Y.$$
\item{\rm d)} If $X$ is metrizable and locally homeomorphic to a
subspace of $\bR^n$, then $\topdim X\le n$.\smallskip
\endclaim

\proof{Proof} a) Apply criterion 13.12~b) and the fact that softness is a local 
property (Prop.\ 4.12).
\medskip
\noindent{b)} When $S$ is closed in $X$, the property follows from Cor.\ 13.10.
When $X$ is metrizable, any subset $S$ is strongly paracompact. Let 
$j:S\longrightarrow X$ be the injection and $\cA$ a sheaf on $S$.
As $\cA=(j_\star\cA)_{\restriction S}$, we have
$$H^q(S,\cA)=H^q(S,j_\star\cA)=
\lim_{\displaystyle\longrightarrow\atop\scriptstyle\Omega\supset S}H^q(\Omega,
j_\star\cA)$$
by Th. 9.10. We may therefore assume that $S$ is open in $X$.
Then every point of $S$ has a neighborhood which is closed in
$X$, so we conclude by a) and the first case of~b).
\medskip
\noindent{c)} Thanks to a) and b), we may assume for example that $X$ is compact.
Let $\cA$ be a sheaf on $X\times Y$ and $\pi:X\times Y\longrightarrow Y$ the
second projection. Set $n_X=\topdim X$, $n_Y=\topdim Y$.
In virtue of $(13.6')$, we have $R^q\pi_\star\cA=0$ for $q>n_X$.
In the Leray spectral sequence, we obtain therefore
$$E^{p,q}_2=H^p(Y,R^q\pi_\star\cA)=0~~~\hbox{\rm for}~~p>n_Y~~\hbox{\rm or}~~
q>n_X,$$
thus $E^{p,l-p}_\infty=0$ when $l>n_X+n_Y$ and we infer
$H^l(X\times Y,\cA)=0$.
\medskip
\noindent{d)} The unit interval $[0,1]\subset\bR$ is of topological dimension 
$\le 1$, because $[0,1]$ admits arbitrarily fine coverings
$$\cU_k=\big(~[0,1]~\cap~](\alpha-1)/k,(\alpha+1)/k[~\big)_{0\le \alpha\le k}
\leqno(13.14)$$
for which only consecutive open sets $U_\alpha$, $U_{\alpha+1}$ intersect;
we may therefore apply Prop.\ 5.24. 
Hence $\bR^n\simeq~]0,1[^n\subset[0,1]^n$ is
of topological dimension $\le n$ by b) and c). Property d) follows.\qed
\endproof

\section{14}{Alexander-Spanier Cohomology}
\subsection{14.A}{Invariance by Homotopy}
Alexander-Spanier's theory can be viewed as the special case of
sheaf cohomology theory with {\it constant coefficients}, i.e.\ with
values in constant sheaves.

\claim{(14.1) Definition} Let $X$ be a topological space,
$R$ a commutative ring and $M$ a $R$-module. 
The constant sheaf $X\times M$ is denoted $M$ for simplicity. 
The Alexander-Spanier $q$-th cohomology group with
values in $M$ is the sheaf cohomology group $H^q(X,M)$.
\endclaim

In particular $H^0(X,M)$ is the set of locally constant functions $X\to M$,
therefore $H^0(X,M)\simeq M^E$, where $E$ is the set of connected components of
$X$. We will not repeat here the properties of Alexander-Spanier
cohomology groups that are formal consequences of those of
general sheaf theory, but we focus our attention instead on new features,
such as invariance by homotopy.

\claim{(14.2) Lemma} Let $I$ denote the interval $[0,1]$ of real numbers.
Then $H^0(I,M)=M$ and $H^q(I,M)=0$ for $q\ne 0$.
\endclaim

\proof{Proof} Let us employ alternate \v Cech cochains for the coverings $\cU_n$
defined in (13.14). As $I$ is paracompact, we have
$$H^q(I,M)=\lim_{\displaystyle\longrightarrow}~~\check H^q(\cU_n,M).$$
However, the alternate \v Cech complex has only two non zero components
and one non zero differential:
$$\eqalign{
&AC^0(\cU_n,M)=\big\{(c_\alpha)_{0\le\alpha\le n}\big\}=M^{n+1},\cr
&AC^1(\cU_n,M)=\big\{(c_{\alpha\,\alpha+1})_{0\le\alpha\le n-1}\big\}=M^n,\cr
&d^0:(c_\alpha)\longmapsto(c'_{\alpha\,\alpha+1})=(c_{\alpha+1}-c_\alpha).
\cr}$$
We see that $d^0$ is surjective, and 
that $\ker d^0=\big\{(m,m,\ldots,m)\big\}=M$.\qed
\endproof

For any continuous map $f:X\longrightarrow Y$, the inverse image of the constant sheaf
$M$ on $Y$ is $f^{-1}M=M$. We get therefore a morphism
$$f^\star: H^q(Y,M)\longrightarrow H^q(X,M),\leqno(14.3)$$
which, as already mentioned in \S 9, is compatible with cup product. 

\claim{(14.4) Proposition} For any space $X$, the projection 
$\pi:X\times I\longrightarrow X$ and the injections $i_t:X\longrightarrow X\times I$,
$x\longmapsto (x,t)$ induce inverse isomorphisms
$$H^q(X,M)~{\raise-4pt\hbox{
$\scriptstyle\pi^\star\atop\displaystyle\relbar\joinrel\longrightarrow$}
\atop\raise4pt\hbox{
$\displaystyle\longleftarrow\joinrel\relbar\atop\scriptstyle i_t^\star$}}
~H^q(X\times I,M).$$
In particular, $i_t^\star$ does not depend on $t$.
\endclaim

\proof{Proof} As $\pi\circ i_t=\Id$, we have $i_t^\star\circ\pi^\star=
\Id$, so it is sufficient to check that $\pi^\star$ is an isomorphism.
However $(R^q\pi_\star M)_x=H^q(I,M)$ in virtue of $(13.6')$, so we get
$$R^0\pi_\star M=M,~~~~R^q\pi_\star M=0~~~\hbox{\rm for}~~q\ne 0$$
and conclude by Cor.\ 13.9.\qed
\endproof

\claim{(14.4) Theorem} If $f,g:X\longrightarrow Y$ are homotopic maps, then
$$f^\star=g^\star:H^q(Y,M)\longrightarrow H^q(X,M).$$
\endclaim

\proof{Proof} Let $H:X\times I\longrightarrow Y$ be a homotopy between $f$ and $g$, with
$f=H\circ i_0$ and $g=H\circ i_1$. Proposition 14.3 implies
$$f^\star=i_0^\star\circ H^\star=i_1^\star\circ H^\star=g^\star.\eqno\square$$
\endproof

We denote $f\sim g$ the homotopy equivalence relation. Two spaces $X,Y$
are said to be homotopically equivalent ($X\sim Y$) if there exist
continuous maps $u:X\longrightarrow Y$, $v:Y\longrightarrow X$ such that $v\circ u\sim\Id_X$
and $u\circ v\sim\Id_Y$. Then $H^q(X,M)\simeq H^q(Y,M)$ and $u^\star,
v^\star$ are inverse isomorphisms.

\claim{(14.5) Example} \rm A subspace $S\subset X$ is said to be a $($strong$)$
deformation retract of $X$ if there exists a {\it retraction} of $X$ onto $S$,
i.e.\ a map $r:X\longrightarrow S$ such that $r\circ j=\Id_S$ ($j=$
inclusion of $S$ in $X$), which is a {\it deformation} of $\Id_X$, i.e.\
there exists a homotopy $H:X\times I\longrightarrow X$ relative to $S$
between $\Id_X$ and $j\circ r$~:
$$H(x,0)=x,~~H(x,1)=r(x)~~\hbox{\rm on}~~X,~~~~H(x,t)=x~~\hbox{\rm on}~~S\times I.$$
Then $X$ and $S$ are homotopically equivalent. In particular $X$ is
said to be {\it contractible} if $X$ has a deformation retraction onto
a point $x_0$. In this case
$$H^q(X,M)=H^q(\{x_0\},M)=\cases{M&for $q=0$\cr 0&for $q\ne 0$.}$$
\endclaim

\claim{(14.6) Corollary} If $X$ is a compact differentiable manifold,
the cohomology groups $H^q(X,R)$ are finitely generated over $R$.
\endclaim

\proof{Proof} Lemma 6.9 shows that $X$ has a finite covering $\cU$ such that
the intersections $U_{\alpha_0\ldots\alpha_q}$ are contractible. Hence
$\cU$ is acyclic, $H^q(X,R)=H^q\big(C^\bu(\cU,R)\big)$ and
each \v Cech cochain space is a finitely generated (free) module.\qed
\endproof

\claim{(14.7) Example: Cohomology Groups of Spheres} \rm Set
$$S^n=\big\{x\in\bR^{n+1}~;~x_0^2+x_1^2+\ldots+x_n^2=1\big\},~~~n\ge 1.$$
We will prove by induction on $n$ that
$$H^q(S^n,M)=\cases{M&for $q=0$ or $q=n$\cr 0&otherwise.\cr}\leqno(14.8)$$
As $S^n$ is connected, we have $H^0(S^n,M)=M$. In order to compute the
higher cohomology groups, we apply the Mayer-Vietoris exact sequence 3.11
to the covering $(U_1,U_2)$ with
$$U_1=S^n\ssm\{(-1,0,\ldots,0)\},~~~~U_2=S^n\ssm\{(1,0,\ldots,0)\}.$$
Then $U_1,U_2\approx\bR^n$ are contractible, and $U_1\cap U_2$ can be
retracted by deformation on the equator $S^n\cap\{x_0=0\}\approx S^{n-1}$.
Omitting $M$ in the notations of cohomology groups, we get exact sequences
$$\leqalignno{&~~~~~~
H^0(U_1)\oplus H^0(U_2)\longrightarrow H^0(U_1\cap U_2)\longrightarrow H^1(S^n)\longrightarrow 0,&(14.9')\cr
&~~~~~~0\longrightarrow H^{q-1}(U_1\cap U_2)\longrightarrow H^q(S^n)\longrightarrow 0,~~~q\ge 2.
&(14.9'')\cr}$$
For $n=1$, $U_1\cap U_2$ consists of two open arcs, so we have
$$H^0(U_1)\oplus H^0(U_2)= H^0(U_1\cap U_2)=M\times M,$$
and the first arrow in $(14.9')$ is $(m_1,m_2)\longmapsto(m_2-m_1,m_2-m_1)$.
We infer easily that $H^1(S^1)=M$ and that 
$$H^q(S^1)=H^{q-1}(U_1\cap U_2)=0~~~\hbox{\rm for}~~q\ge 2.$$
For $n\ge 2$, $U_1\cap U_2$ is connected, so the first arrow in $(14.9')$ 
is onto and $H^1(S^n)=0$. The second sequence $(14.9'')$ yields
$H^q(S^n)\simeq H^{q-1}(S^{n-1})$.
An induction concludes the proof.\qed
\endclaim

\subsection{14.B}{Relative Cohomology Groups and Excision Theorem}
Let $X$ be a topological space and $S$ a subspace. We denote by $M^{[q]}(X,S)$
the subgroup of sections $u\in M^{[q]}(X)$ such that $u(x_0,\ldots,x_q)=0$ when
$$(x_0,\ldots,x_q)\in S^q,~~~x_1\in V(x_0),~\ldots,~x_q\in V(x_0,\ldots,x_{q-1}).
$$
Then $M^{[\bu]}(X,S)$ is a subcomplex of $M^{[\bu]}(X)$
and we define the {\it relative cohomology group} of the pair $(X,S)$ with 
values in $M$ as
$$H^q(X,S\,;\,M)=H^q\big(M^{[\bu]}(X,S)\big).\leqno(14.10)$$
By definition of $M^{[q]}(X,S)$, there is an exact sequence
$$0\longrightarrow M^{[q]}(X,S)\longrightarrow M^{[q]}(X)\longrightarrow(M_{\restriction S})^{[q]}(S)\longrightarrow 0.
\leqno(14.11)$$
The reader should take care of the fact that $(M_{\restriction S})^{[q]}(S)$
does not coincide with the module of sections $M^{[q]}(S)$ of the sheaf 
$M^{[q]}$ on $X$, except if $S$ is open. The snake lemma shows that there 
is an ``exact sequence of the pair":
$$(14.12)~~H^q(X,S\,;\,M)\to H^q(X,M)\to H^q(S,M)\to H^{q+1}(X,S\,;\,M)
\cdots.$$
We have in particular $H^0(X,S\,;\,M)=M^E$, where $E$ is the set of
connected components of $X$ which do not meet $S$.
More generally, for a triple $(X,S,T)$ with $X\supset S\supset T$, there is
an ``exact sequence of the triple":
$$\leqalignno{
0&\longrightarrow M^{[q]}(X,S)\longrightarrow M^{[q]}(X,T)\longrightarrow M^{[q]}(S,T)\longrightarrow 0,&(14.12')\cr
~~~~H^q(X,S\,;\,&M)\longrightarrow H^q(X,T\,;\,M)\longrightarrow H^q(S,T\,;\,M)\longrightarrow
H^{q+1}(X,S\,;\,M).\cr}$$
The definition of the cup product in (8.2) shows that $\alpha\smallsmile
\beta$ vanishes on $S\cup S'$ if $\alpha,\beta$ vanish on $S$, $S'$ 
respectively. Therefore, we obtain a bilinear map
$$H^p(X,S\,;\,M)\times H^q(X,S'\,;\,M')\longrightarrow
H^{p+q}(X,S\cup S'\,;\,M\otimes M').\leqno(14.13)$$
If $f:(X,S)\longrightarrow(Y,T)$ is a morphism of pairs, i.e.\ a continuous map $X\to Y$
such that $f(S)\subset T$, there is an induced pull-back morphism
$$f^\star:~~H^q(Y,T\,;\,M)\longrightarrow H^q(X,S\,;\,M)\leqno(14.14)$$
which is compatible with the cup product. Two morphisms of pairs
$f,g$ are said to be homotopic when there is a pair homotopy
$H:(X\times I,S\times I)\longrightarrow(Y,T)$. An application of the exact sequence
of the pair shows that 
$$\pi^\star:H^q(X,S\,;\,M)\longrightarrow H^q(X\times I,S\times I\,;\,M)$$
is an isomorphism. It follows as above that $f^\star=g^\star$ as soon
as $f,g$ are homotopic.

\claim{(14.15) Excision theorem} For subspaces
$\ol T\subset S^\circ$ of $X$,
the restriction morphism 
$H^q(X,S\,;\,M)\longrightarrow H^q(X\ssm T,S\ssm T\,;\,M)$
is an isomorphism.
\endclaim

\proof{Proof} Under our assumption, it is not difficult to check that the 
surjective restriction map $M^{[q]}(X,S)\longrightarrow M^{[q]}(X\ssm T,
S\ssm T)$ is also injective, because the kernel consists
of sections $u\in M^{[q]}(X)$ such that $u(x_0,\ldots,x_q)=0$
on $(X\ssm T)^{q+1}\cup S^{q+1}$, and this set is a neighborhood
of the diagonal of $X^{q+1}$.\qed
\endproof

\claim{(14.16) Proposition} If $S$ is open or strongly paracompact
in $X$, the relative cohomology groups can be written in terms of
cohomology groups with supports in $X\ssm S\,$:
$$H^q(X,S\,;\,M)\simeq H^q_{X\ssm S}(X,M).$$
In particular, if $X\ssm S$ is relatively compact in $X$, we have
$$H^q(X,S\,;\,M)\simeq H^q_c(X\ssm S,M).$$
\endclaim

\proof{Proof} We have an exact sequence
$$0\longrightarrow M^{[\bu]}_{X\ssm S}(X)\longrightarrow M^{[\bu]}(X)\longrightarrow
M^{[\bu]}(S)\longrightarrow 0\leqno(14.17)$$
where $M^{[\bu]}_{X\ssm S}(X)$ denotes sections with support
in $X\ssm S$.
If $S$ is open, then $M^{[\bu]}(S)=(M_{\restriction S})^{[\bu]}(S)$,
hence $M^{[\bu]}_{X\ssm S}(X)=M^{[\bu]}(X,S)$
and the result follows. If $S$ is strongly paracompact, Prop.\ 4.7 
and Th. 9.10 show that
$$H^q\big(M^{[\bu]}(S)\big)=H^q\big(
\lim_{\displaystyle\longrightarrow\atop\scriptstyle\Omega\supset S}M^{[\bu]}(\Omega)\big)
=\lim_{\displaystyle\longrightarrow\atop\scriptstyle\Omega\supset S}H^q(\Omega,M)
=H^q(S,M_{\restriction S}).$$
If we consider the diagram
$$\cmalign{
0&~\longrightarrow~&M^{[\bu]}_{X\ssm S}(X)&~\longrightarrow~&M^{[\bu]}(X)&~\longrightarrow~
&M^{[\bu]}(S)&~\longrightarrow~ 0\cr
\noalign{\vskip4pt}
&&~~~~~~\big\downarrow&&~~~\big\downarrow \Id&&~~~\big\downarrow
\restriction S&\cr
\noalign{\vskip4pt}
0&~\longrightarrow~&M^{[\bu]}(X,S)&~\longrightarrow~&M^{[\bu]}(X)&~\longrightarrow~
&(M_{\restriction S})^{[\bu]}(S)&~\longrightarrow~ 0\cr}$$
we see that the last two vertical arrows induce isomorphisms in cohomology.
Therefore, the first one also does.\qed
\endproof

\claim{(14.18) Corollary} Let $X,Y$ be locally compact spaces and
\hbox{$f,g:X\to Y$} proper maps. We say that $f,g$ are properly homotopic
if they are homotopic through a proper homotopy $H:X\times I\longrightarrow Y$. Then
$$f^\star=g^\star~:~~H^q_c(Y,M)\longrightarrow H^q_c(X,M).$$
\endclaim

\proof{Proof} Let $\wh X=X\cup\{\infty\}$, $\wh Y=Y\cup\{\infty\}$ be the Alexandrov
compactifications of $X$, $Y$. Then $f,g,H$ can be extended as
continuous maps
$$\wh f,\wh g~:~~\wh X\longrightarrow\wh Y,~~~~\wh H~:~~\wh X\times I\longrightarrow\wh Y$$
with $\wh f(\infty)=\wh g(\infty)=H(\infty,t)=\infty$, so that
$\wh f,\wh g$ are homotopic as maps $(\wh X,\infty)\longrightarrow (\wh Y,\infty)$.
Proposition 14.16 implies $H^q_c(X,M)=H^q(\wt X,\infty\,;\,M)$
and the result follows.\qed
\endproof

\section{15}{K\"unneth Formula}
\subsection{15.A}{Flat Modules and Tor Functors}
The goal of this section is to investigate homological properties
related to tensor products. We work in the category of modules 
over a commutative ring $R$ with unit. All tensor products appearing 
here are tensor products over $R$. The starting point is the observation
that tensor product
with a given module is a right exact functor: if $0\to A\to B\to C\to 0$
is an exact sequence and $M$ a $R$-module, then
$$A\otimes M\longrightarrow B\otimes M\longrightarrow C\otimes M\longrightarrow 0$$
is exact, but the map $A\otimes M\longrightarrow B\otimes M$ need not be injective.
A counterexample is given by the sequence
$$0\longrightarrow\bZ\buildo 2\times\over\longrightarrow\bZ\longrightarrow\bZ/2\bZ\longrightarrow0~~~~\hbox{\rm over}~~R=\bZ$$
tensorized by $M=\bZ/2\bZ$. However, the injectivity holds if $M$ is a
free $R$-module. More generally, one says that $M$ is a {\it flat
$R$-module} if the tensor product by $M$ preserves injectivity,
or equivalently, if $\otimes M$ is a left exact functor.

A {\it flat resolution} $C_\bu$ of a $R$-module $A$ is a homology
exact sequence
$$\cdots\longrightarrow C_q\longrightarrow C_{q-1}\longrightarrow\cdots\longrightarrow C_1\longrightarrow C_0\longrightarrow A\longrightarrow 0$$
where $C_q$ are flat $R$-modules and $C_q=0$ for $q<0$. Such a resolution
always exists because every module $A$ is a quotient of a free module
$C_0$. Inductively, we take $C_{q+1}$ to be a free module such that
$\ker(C_q\to C_{q-1})$ is a quotient of $C_{q+1}$.
In terms of homology groups, we have $H_0(C_\bu)=A$ and $H_q(C_\bu)=0$
for $q\ne 0$. Given $R$-modules $A,B$ and free resolutions
$d':C_\bu\longrightarrow A$, $d'':D_\bu\longrightarrow B$, we consider the double homology complex
$$K_{p,q}=C_p\otimes D_q,~~~~d=d'\otimes\Id+(-1)^p\Id\otimes d''$$
and the associated first and second spectral sequences. Since
$C_p$ is free, we have
$$E^1_{p,q}=H_q(C_p\otimes D_\bu)=\cases{C_p\otimes B&for~~$q=0$,\cr
0&for~~$q\ne0$.\cr}$$
Similarly, the second spectral sequence also collapses and we have
$$H_l(K_\bu)=H_l(C_\bu\otimes B)=H_l(A\otimes D_\bu).$$
This implies in particular that the homology groups $H_l(K_\bu)$ do not 
depend on the choice of the resolutions $C_\bu$ or $D_\bu$.

\claim{(15.1) Definition} The $q$-th torsion module of $A$ and $B$ is
$$\Tor_q(A,B)=H_q(K_\bu)=H_q(C_\bu\otimes B)=H_q(A\otimes D_\bu).$$
\endclaim

Since the definition of $K_\bu$ is symmetric with respect to $A$, $B$,
we have $\Tor_q(A,B)\simeq\Tor_q(B,A)$. By the right-exactness of 
$\otimes B$, we find in particular $\Tor_0(A,B)=A\otimes B$. 
Moreover, if $B$ is flat, $\otimes B$ is also left exact, thus 
$\Tor_q(A,B)=0$ for all $q\ge 1$ and all modules $A$. If 
$0\to A\to A'\to A''\to 0$ is an exact sequence,
there is a corresponding exact sequence of homology complexes
$$0\longrightarrow A\otimes D_\bu\longrightarrow A'\otimes D_\bu\longrightarrow A''\otimes D_\bu\longrightarrow 0,$$
thus a long exact sequence
$$\cmalign{
&\longrightarrow\Tor_q(A,B)&\longrightarrow\Tor_q(A',B)&\longrightarrow\Tor_q(A'',B)&\longrightarrow
\Tor_{q-1}(A,B)\cr
\hfill\cdots&\longrightarrow~~~A\otimes B&\longrightarrow~~~A'\otimes B&\longrightarrow~~~A''\otimes B
&\longrightarrow~~~0.\cr}\leqno(15.2)$$
It follows that $B$ is flat if and only if $\Tor_1(A,B)=0$ for
every $R$-module $A$.

Suppose now that $R$ is a {\it principal ring}. Then
every module $A$ has a free resolution $0\to C_1\to C_0\to A\to 0$
because the kernel of any surjective map $C_0\to A$ is free
(every submodule of a free module is free). It follows
that one always has $\Tor_q(A,B)=0$ for $q\ge 2$. In this case,
we denote $\Tor_1(A,B)=A\star B$ and call it the {\it torsion product}
of $A$ and $B$. The above exact sequence (15.2) reduces to
$$0\to A\star B\to A'\star B\to A''\star B\to A\otimes B\to A'\otimes B\to
A''\otimes B\to 0.\leqno(15.3)$$
In order to compute $A\star B$, we may restrict
ourselves to finitely generated modules, because every module is
a direct limit of such modules and the $\star$ product commutes
with direct limits. Over a principal ring $R$,
every finitely generated module is a direct sum of a free module
and of cyclic modules $R/aR$. It is thus sufficient to compute
$R/aR\star R/bR$. The obvious free resolution
$\smash{R\buildo a\times\over\longrightarrow R}$ of $R/aR$ shows that $R/aR\star R/bR$
is  the kernel of the map $\smash{R/bR\buildo a\times\over\longrightarrow R/bR}$. Hence
$$R/aR\star R/bR\simeq R/(a\wedge b)R\leqno(15.4)$$
where $a\wedge b$ denotes the greatest common divisor of $a$ and $b$.
It follows that a module $B$ is flat if and only if it is torsion free.
If $R$ is a field, every $R$-module is free, thus $A\star B=0$ for
all $A$ and $B$.

\subsection{15.B}{K\"unneth and Universal Coefficient Formulas}
The algebraic K\"unneth formula describes the cohomology groups of the
tensor product of two differential complexes.

\claim{(15.5) Algebraic K\"unneth formula} Let $(K^\bu,d')$,
$(L^\bu,d'')$ be complexes of $R$-modules and
$(K\otimes L)^\bu$ the simple complex associated to the double complex
$(K\otimes L)^{p,q}=K^p\otimes L^q$. If $K^\bu$ or $L^\bu$ is torsion free,
there is a split exact sequence
$$0\to\bigoplus_{p+q=l}H^p(K^\bu)\otimes H^q(L^\bu)\buildo\mu\over\to 
H^l\big((K\otimes L)^\bu\big)\to\bigoplus_{p+q=l+1}H^p(K^\bu)\star
\buildu{\displaystyle\to 0\phantom{\Big|}}\under{H^q(L^\bu)}$$
where the map $\mu$ is defined by $\mu(\{k^p\}\times\{l^q\})=\{k^p\otimes
l^q\}$ for all cocycles $\{k^p\}\in H^p(K^\bu)$, $\{l^q\}\in H^q(L^\bu)$.
\endclaim

\claim{(15.6) Corollary} If $R$ is a field, or if one of the graded modules
$H^\bu(K^\bu)$, $H^\bu(L^\bu)$ is torsion free, then
$$H^l\big((K\otimes L)^\bu\big)\simeq\bigoplus_{p+q=l}H^p(K^\bu)\otimes
H^q(L^\bu).$$
\endclaim

\proof{Proof} Assume for example that $K^\bu$ is torsion free.
There is a short exact sequence of complexes
$$0\longrightarrow Z^\bu\longrightarrow K^\bu\buildo d'\over\longrightarrow B^{\bu+1}\longrightarrow 0$$
where $Z^\bu,B^\bu\subset K^\bu$ are respectively the graded modules of
cocycles and coboundaries in $K^\bu$, considered as subcomplexes with
zero differential. As $B^{\bu+1}$ is torsion free, the tensor product of
the above sequence with $L^\bu$ is still exact. The corresponding long exact 
sequence for the associated simple complexes yields:
$$\leqalignno{
H^l\big((B\otimes L)^\bu\big)&\longrightarrow H^l\big((Z\otimes L)^\bu\big)\longrightarrow
H^l\big((K\otimes L)^\bu\big)\buildo d'\over\longrightarrow H^{l+1}
\big((B\otimes L)^\bu\big)\cr
&\longrightarrow H^{l+1}\big((Z\otimes L)^\bu\big)\cdots.&(15.7)\cr}$$
The first and last arrows are connecting homomorphisms; in this situation, 
they are easily seen to be induced by the inclusion $B^\bu\subset Z^\bu$.
Since the differential of $Z^\bu$ is zero, the simple complex 
$(Z\otimes L)^\bu$ is isomorphic to the direct sum $\bigoplus_p Z^p\otimes
L^{\bu-p}$, where $Z^p$ is torsion free. Similar properties hold for 
$(B\otimes L)^\bu$, hence
$$H^l\big((Z\otimes L)^\bu\big)=\bigoplus_{p+q=l}Z^p\otimes H^q(L^\bu),~~~~
H^l\big((B\otimes L)^\bu\big)=\bigoplus_{p+q=l}B^p\otimes H^q(L^\bu).$$
The exact sequence
$$0\longrightarrow B^p\longrightarrow Z^p\longrightarrow H^p(K^\bu)\longrightarrow 0$$
tensorized by $H^q(L^\bu)$ yields an exact sequence of the type (15.3):
$$\eqalign{
0\to H^p(K^\bu)\star H^q(L^\bu)\to B^p\otimes&H^q(L^\bu)\to Z^p\otimes
H^q(L^\bu)\cr
&\to H^p(K^\bu)\otimes H^q(L^\bu)\to 0.\cr}$$
By the above equalities, we get
$$\eqalign{
0\longrightarrow\bigoplus_{p+q=l}H^p(K^\bu)\star H^q&(L^\bu)\longrightarrow
H^l\big((B\otimes L)^\bu\big)\longrightarrow H^l\big((Z\otimes L)^\bu\big)\cr
&\longrightarrow\bigoplus_{p+q=l}H^p(K^\bu)\otimes H^q(L^\bu)\longrightarrow 0.\cr}$$
In our initial long exact sequence (15.7), the cokernel of the first
arrow is thus 
$$
\bigoplus_{p+q=l}H^p(K^\bu)\otimes H^q(L^\bu)
$$
and the kernel of the last arrow is the torsion sum
$\bigoplus_{p+q=l+1}H^p(K^\bu)\star H^q(L^\bu)$.
This gives the exact sequence of the lemma. We leave the computation of the
map $\mu$ as an exercise for the reader. The splitting assertion
can be obtained by observing that there always exists a torsion free complex 
$\wt K^\bu$ that splits (i.e.\ $\wt Z^\bu\subset\wt K^\bu$ splits),
and a morphism $\wt K^\bu\longrightarrow K^\bu$ inducing an isomorphism
in cohomology; then the projection $\wt K^\bu\longrightarrow\wt Z^\bu$ yields a
projection
$$\eqalign{
H^l\big((\wt K\otimes L)^\bu\big)\longrightarrow H^l\big((\wt Z\otimes L)^\bu\big)
\simeq\bigoplus_{p+q=l}\wt Z^p&\otimes H^q(L^\bu)\cr
&\longrightarrow\bigoplus_{p+q=l}H^p(\wt K^\bu)\otimes H^q(L^\bu).\cr}$$
To construct $\wt K^\bu$, let $\wt Z^\bu\longrightarrow Z^\bu$ be a surjective map
with $\wt Z^\bu$ free, $\wt B^\bu$ the inverse image of $B^\bu$ in 
$\wt\cZ^\bu$ and $\wt K^\bu=\wt Z^\bu\oplus\wt B^{\bu+1}$, where the 
differential $\wt K^\bu\longrightarrow \wt K^{\bu+1}$ is given by $\wt Z^\bu\longrightarrow 0$
and $\wt B^{\bu+1}\subset\wt Z^{\bu+1}\oplus 0$~;
as $\wt B^\bu$ is free, the map $\wt B^{\bu+1}\longrightarrow B^{\bu+1}$ can be lifted
to a map $\wt B^{\bu+1}\longrightarrow K^\bu$, and this lifting combined with the composite
$\wt Z^\bu\to Z^\bu\subset K^\bu$ yields the required complex morphism 
$\wt K^\bu=\wt Z^\bu\oplus\wt B^{\bu+1}\longrightarrow K^\bu$.\qed
\endproof

\claim{(15.8) Universal coefficient formula} Let $K^\bu$ be a
complex of $R$-modules and $M$ a $R$-module such that either 
$K^\bu$ or $M$ is torsion free. Then there is a split exact sequence
$$0\longrightarrow H^p(K^\bu)\otimes M\longrightarrow H^p(K^\bu\otimes M)\longrightarrow H^{p+1}(K^\bu)\star M
\longrightarrow 0.$$
\endclaim

Indeed, this is a special case of Formula~15.5 when the complex $L^\bu$
is reduced to one term $L^0=M$. In general, it is interesting to observe
that the spectral sequence of $K^\bu\otimes L^\bu$ collapses in $E_2$ if
$K^\bu$ is torsion free: $H^k\big((K\otimes L)^\bu\big)$ is in
fact the direct sum of the terms $E^{p,q}_2=H^p\big(K^\bu\otimes
H^q(L^\bu)\big)$ thanks to (15.8).

\subsection{15.C}{K\"unneth Formula for Sheaf Cohomology}
Here we apply the general algebraic machinery to compute cohomology
groups over a product space $X\times Y$. The main argument is a
combination of the Leray spectral sequence with
the universal coefficient formula for sheaf cohomology.

\claim{(15.9) Theorem} Let $\cA$ be a sheaf of $R$-modules over a
topological space $X$ and $M$ a $R$-module. Assume that either $\cA$ 
or $M$ is torsion free and that either $X$ is compact or $M$ is finitely
generated. Then there is a split exact sequence
$$0\longrightarrow H^p(X,\cA)\otimes M\longrightarrow H^p(X,\cA\otimes M)\longrightarrow H^{p+1}(X,\cA)\star M
\longrightarrow 0.$$
\endclaim

\proof{Proof} If $M$ is finitely generated, we get
$(\cA\otimes M)^{[\bu]}(X)=\cA^{[\bu]}(X)\otimes M$ easily, so the above
exact sequence is a consequence of Formula~15.8.
If $X$ is compact, we may consider \v Cech cochains $C^q(\cU,\cA\otimes M)$
over finite coverings. There is an obvious morphism
$$C^q(\cU,\cA)\otimes M\longrightarrow C^q(\cU,\cA\otimes M)$$
but this morphism need not be surjective nor injective. However, since
$$(\cA\otimes M)_x=\cA_x\otimes M=
\lim_{\displaystyle\longrightarrow\atop\scriptstyle V\ni x}~~\cA(V)\otimes M,$$
the following properties are easy to verify:
\medskip
\item{a)} If $c\in C^q(\cU,\cA\otimes M)$, there is a refinement
$\cV$ of $\cU$ and $\rho:\cV\longrightarrow\cU$ such that
$\rho^\star c\in C^q(\cV,\cA\otimes M)$ is in
the image of $C^q(\cV,\cA)\otimes M$.
\medskip
\item{b)} If a tensor $t\in C^q(\cU,\cA)\otimes M$ is mapped to $0$
in $C^q(\cU,\cA\otimes M)$, there is a refinement
$\cV$ of $\cU$ such that $\rho^\star t\in C^q(\cV,\cA)\otimes M$ equals $0$.
\medskip
\noindent From a) and b) it follows that
$$\check H^q(X,\cA\otimes M)=
\lim_{\displaystyle\longrightarrow\atop\scriptstyle\cU}~~
H^q\big(C^\bu(\cU,\cA\otimes M)\big)
=\lim_{\displaystyle\longrightarrow\atop\scriptstyle\cU}~~
H^q\big(C^\bu(\cU,\cA)\otimes M\big)$$
and the desired exact sequence is the direct limit of the exact
sequences of Formula~15.8 obtained for $K^\bu=C^\bu(\cU,\cA)$.\qed
\endproof

\claim{(15.10) Theorem {\rm(K\"unneth)}} Let $\cA$ and $\cB$ 
be sheaves of $R$-modules over topological spaces $X$ and $Y$. Assume that
$\cA$ is torsion free, that $Y$ is compact and that either $X$ is
compact or the cohomology groups $H^q(Y,\cB)$ are finitely generated
$R$-modules. There is a split exact sequence
$$\eqalign{
0\longrightarrow\bigoplus_{p+q=l}H^p(X,\cA)&\otimes H^q(Y,\cB)\buildo\mu\over\longrightarrow
H^l(X\times Y,\cA\stimes\cB)\cr
&\longrightarrow\bigoplus_{p+q=l+1}H^p(X,\cA)\star H^q(Y,\cB)\longrightarrow 0\cr}$$
where $\mu$ is the map given by the cartesian product
$\bigoplus\alpha_p\otimes\beta_q\longmapsto\sum \alpha_p\times\beta_q.$
\endclaim

\proof{Proof} We compute $H^l(X,\cA\stimes\cB)$ by means of the Leray spectral
sequence of the projection $\pi:X\times Y\longrightarrow X$. This means that we are
considering the differential sheaf $\cL^q=\pi_\star(\cA\stimes\cB)^{[q]}$
and the double complex
$$K^{p,q}=(\cL^q)^{[p]}(X).$$
By $(12.5')$ we have ${}_K E_2^{p,q}=H^p\big(X,\cH^q(\cL^\bu)\big)$.
As $Y$ is compact, the cohomology sheaves 
$\cH^q(\cL^\bu)=R^q\pi_\star(\cA\stimes\cB)$ are given by
$$R^q\pi_\star(\cA\stimes\cB)_x\!=\!H^q(\{x\}\times Y,\cA\stimes
\cB_{\restriction\{x\}\times Y})\!=\!H^q(Y,\cA_x\otimes\cB)\!=\!
\cA_x\otimes H^q(Y,\cB)$$
thanks to the compact case of Th. 15.9 where $M=\cA_x$ is torsion
free. We obtain therefore
$$\eqalign{
&R^q\pi_\star(\cA\stimes\cB)=\cA\otimes H^q(Y,\cB),\cr
&{}_KE^{p,q}_2=H^p\big(X,\cA\otimes H^q(Y,\cB)\big).\cr}$$
Theorem 15.9 shows that the $E_2$-term is actually given by the
desired exact sequence, but it is not a priori clear that the spectral
sequence collapses in $E_2$. In order to check this, we consider the
double complex
$$C^{p,q}=\cA^{[p]}(X)\otimes\cB^{[q]}(Y)$$
and construct a natural morphism $C^{\bu,\bu}\longrightarrow K^{\bu,\bu}$. We
may consider the set $K^{p,q}=\big(\pi_\star(\cA\stimes\cB)^{[q]}\big)^{[p]}(X)$
as the set of equivalence classes of functions
$$h\big(\xi_0,\ldots,\xi_p)\in\pi_\star(\cA\stimes\cB)^{[q]}_{\xi_p}
=\lim_{\displaystyle\longrightarrow}~~(\cA\stimes\cB)^{[q]}\big(\pi^{-1}
\big(V(\xi_p)\big)\big)$$
or more precisely
$$\eqalign{
&h\big(\xi_0,\ldots,\xi_p\,;\,(x_0,y_0),\ldots,(x_q,y_q)\big)\in\cA_{x_q}\otimes\cB_{y_q}
~~~~\hbox{\rm with}\cr
&\xi_0\in X,~~~\xi_j\in V(\xi_0,\ldots,\xi_{j-1}),~~~1\le j\le p,\cr
&(x_0,y_0)\in V(\xi_0,\ldots,\xi_p)\times Y,\cr
&(x_j,y_j)\in V\big(\xi_0,\ldots,\xi_p\,;\,(x_0,y_0),\ldots,(x_{j-1},y_{j-1})\big),
~~~1\le j\le q.\cr}$$
Then $f\otimes g\in C^{p,q}$ is mapped to $h\in K^{p,q}$ by the formula
$$h\big(\xi_0,\ldots,\xi_p\,;\,(x_0,y_0),\ldots,(x_q,y_q)\big)=
f(\xi_0,\ldots,\xi_p)(x_q)\otimes g(y_0,\ldots,y_q).$$
As $\cA^{[p]}(X)$ is torsion free, we find
$${}_CE^{p,q}_1=\cA^{[p]}(X)\otimes H^q(Y,\cB).$$
Since either $X$ is compact or $H^q(Y,\cB)$ finitely generated, Th.
15.9 yields
$${}_CE^{p,q}_2=H^p\big(X,\cA\otimes H^q(Y,\cB)\big)\simeq{}_KE^{p,q}_2$$
hence $H^l(K^\bu)\simeq H^l(C^\bu)$ and the algebraic K\"unneth formula 
15.5 concludes the proof.\qed
\endproof

\claim{(15.11) Remark} \rm The exact sequences of Th. 15.9 and of
K\"unneth's theorem also hold for cohomology groups with compact
support, provided that $X$ and $Y$ are locally compact and
$\cA$ (or $\cB$) is torsion free. This is an immediate consequence
of Prop.\ 7.12 on direct limits of cohomology groups
over compact subsets.
\endclaim

\claim{(15.12) Corollary} When $\cA$ and $\cB$ are torsion free
constant sheaves, e.g.\ $\cA=\cB=\bZ$ or $\bR$, the K\"unneth formula
holds as soon as $X$ or $Y$ has the same homotopy type as a finite
cell complex.
\endclaim

\proof{Proof} If $Y$ satisfies the assumption, we may suppose in fact that $Y$
is a finite cell complex by the homotopy invariance. Then $Y$ is compact
and $H^\bu(Y,\cB)$ is finitely generated, so Th.~15.10 can be applied.\qed
\endproof

\section{16}{Poincar\'e duality}
\subsection{16.A}{Injective Modules and Ext Functors}
The study of duality requires some algebraic preliminaries on the ~Hom~ 
functor and its derived functors $\Ext^q$. Let $R$ be a commutative
ring with unit, $M$ a $R$-module and
$$0\longrightarrow A\longrightarrow B\longrightarrow C\longrightarrow 0$$
an exact sequence of $R$-modules. Then we have exact sequences
$$\eqalign{
0\longrightarrow&\Hom_R(M,A)\longrightarrow\Hom_R(M,B)\longrightarrow\Hom_R(M,C),\cr
&\Hom_R(A,M)\longleftarrow\Hom_R(B,M)\longleftarrow\Hom_R(C,M)
\longleftarrow 0,\cr}$$
i.e.\ $\Hom(M,\bu)$ is a covariant left exact functor and $\Hom(\bu,M)$ a
contravariant right exact functor. The module $M$ is said to be
{\it projective} if $\Hom(M,\bu)$ is also right exact, and {\it injective}
if $\Hom(\bu,M)$ is also left exact. Every free $R$-module is projective.
Conversely, if $M$ is projective, any surjective morphism $F\longrightarrow M$ from
a free module $F$ onto $M$ must split $\big(\Id_M$ has a preimage in
$\Hom(M,F)\big)$; if $R$ is a principal ring,
``projective" is therefore equivalent to ``free".

\claim{(16.1) Proposition} Over a principal ring $R$, a module $M$
is injective if and only if it is divisible, i.e.\ if for every $x\in M$
and $\lambda\in R\ssm\{0\}$, there exists $y\in M$ such that
$\lambda y=x$.
\endclaim

\proof{Proof} If $M$ is injective, the exact sequence
$0\longrightarrow R\buildo\lambda\times\over\longrightarrow R\longrightarrow R/\lambda R\longrightarrow 0$ shows that
$$M=\Hom(R,M)\buildo\lambda\times\over\longrightarrow\Hom(R,M)=M$$
must be surjective, thus $M$ is divisible.

Conversely, assume that $R$ is divisible. Let $f:A\longrightarrow M$ be a morphism
and $B\supset A$. Zorn's lemma implies that there is a maximal
extension $\smash{\wt f:\wt A}\longrightarrow M$ of $f$ where
$A\subset\smash{\wt A}\subset B$. If $\smash{\wt A}\ne B$, select 
$x\in B\ssm\smash{\wt A}$ and consider the ideal $I$ of elements
$\lambda\in R$ such that $\lambda x\in\smash{\wt A}$. As $R$ is
principal we have $I=\lambda_0R$ for some $\lambda_0$.
If $\lambda_0\ne 0$, select $y\in M$ such that $\lambda_0y=\smash{\wt f}
(\lambda_0x)$ and if $\lambda_0=0$ take $y$ arbitrary. Then
$\wt f$ can be extended to $\smash{\wt A}+Rx$ by letting $\smash{\wt f}(x)=y$.
This is a contradiction, so we must have $\smash{\wt A}=B$.\qed
\endproof

\claim{(16.2) Proposition} Every module $M$ can be embedded in an
injective \hbox{module $\smash{\wt M}$.}
\endclaim

\proof{Proof} Assume first $R=\bZ$. Then set
$$M'=\Hom_\bZ(M,\bQ/\bZ),~~~~M''=\Hom_\bZ(M',\bQ/\bZ)\subset\bQ/\bZ^{M'}.$$
Since $\bQ/\bZ$ is divisible, $\bQ/\bZ$ and $\bQ/\bZ^{M'}$ are
injective.
It is therefore sufficient to show that the canonical morphism
$M\longrightarrow M''$ is injective. In fact, for any $x\in M\ssm\{0\}$, the
subgroup $\bZ x$ is cyclic (finite or infinite), so there is a non
trivial morphism $\bZ x\longrightarrow\bQ/\bZ$, and we can extend this morphism
into a morphism $u:M\longrightarrow\bQ/\bZ$. Then $u\in M'$ and $u(x)\ne 0$, so
$M\longrightarrow M''$ is injective.

Now, for an arbitrary ring $R$, we set $\wt M=\Hom_\bZ\big(R,
\bQ/\bZ^{M'}\big)$. There are $R$-linear embeddings
$$M=\Hom_R(R,M)\lhra\Hom_\bZ(R,M)\lhra\Hom_\bZ\big(R,\bQ/\bZ^{M'}\big)
=\wt M$$
and since $\Hom_R(\bu,\wt M)\simeq\Hom_\bZ\big(\bu,\bQ/\bZ^{M'}\big)$, it
is clear that $\smash{\wt M}$ is injective over the ring~$R$.\qed
\endproof

As a consequence, any module has a
(cohomological) resolution by injective modules. Let $A,B$ be given
$R$-modules, let $d':B\to D^\bu$ be an injective resolution of $B$
and let $d'':C_\bu\to A$ be a free (or projective) resolution of~$A$.
We consider the cohomology double complex
$$K^{p,q}=\Hom(C_q,D^p),~~~~d=d'+(-1)^p(d'')^\dagger$$
($\dagger$ means transposition) and the associated first and second 
spectral sequences. Since $\Hom(\bu,D^p)$ and $\Hom(C_q,\bu)$ are exact,
we get
$$\eqalign{
E^{p,0}_1&=\Hom(A,D^p),~~~~\wt E^{p,0}_1=\Hom(C_p,B),\cr
E^{p,q}_1&=\wt E^{p,q}_1=0~~~~\hbox{\rm for}~~q\ne 0.\cr}$$
Therefore, both spectral sequences collapse in $E_1$ and we get
$$H^l(K^\bu)=H^l\big(\Hom(A,D^\bu)\big)=H^l\big(\Hom(C_\bu,B)\big)~;$$
in particular, the cohomology groups $H^l(K^\bu)$ do not depend on the
choice of the resolutions $C_\bu$ or $D^\bu$.

\claim{(16.3) Definition} The $q$-th extension module of $A$, $B$ is
$$\Ext^q_R(A,B)=H^q(K^\bu)=H^q\big(\Hom(A,D^\bu)\big)
=H^q\big(\Hom(C_\bu,B)\big).$$
\endclaim

By the left exactness of $\Hom(A,\bu)$, we get in particular
$\Ext^0(A,B)=\Hom(A,B)$. If $A$ is projective or $B$ injective,
then clearly $\Ext^q(A,B)=0$ for all $q\ge 1$. Any exact sequence
$0\to A\to A'\to A''\to 0$ is converted into an exact sequence by
$\Hom(\bu,D^\bu)$, thus we get a long exact sequence
$$\eqalign{
0&\longrightarrow\Hom(A'',B)\longrightarrow\Hom(A',B)\longrightarrow\Hom(A,B)\longrightarrow\Ext^1(A'',B)\cdots\cr
&\longrightarrow\Ext^q(A'',B)\longrightarrow\Ext^q(A',B)\longrightarrow\Ext^q(A,B)
 \longrightarrow\Ext^{q+1}(A'',B)\cdots\cr}$$
Similarly, any exact sequence $0\to B\to B'\to B''\to 0$ yields
$$\eqalign{
0&\longrightarrow\Hom(A,B)\longrightarrow\Hom(A,B')\longrightarrow\Hom(A,B'')\longrightarrow\Ext^1(A,B)\cdots\cr
&\longrightarrow\Ext^q(A,B)\longrightarrow\Ext^q(A,B')\longrightarrow\Ext^q(A,B'')
 \longrightarrow\Ext^{q+1}(A,B)\cdots\cr}$$
Suppose now that $R$ is a principal ring. Then the resolutions $C_\bu$
or $D^\bu$ can be taken of length $1$ (any quotient of a divisible
module is divisible), thus $\Ext^q(A,B)$ is always $0$ for
$q\ge 2$. In this case, we simply denote $\Ext^1(A,B)=
\Ext(A,B)$. When $A$ is finitely generated, the computation of
$\Ext(A,B)$ can be reduced to the cyclic case, since
$\Ext(A,B)=0$ when $A$ is free. For $A=R/aR$, the obvious free
resolution $R\smash{\buildo a\times\over\longrightarrow}R$ gives
$$\Ext_R(R/aR,B)=B/aB.\leqno(16.4)$$

\claim{(16.5) Lemma} Let $K_\bu$ be a homology complex and let $M\to M^\bu$ be
an injective resolution of a $R$-module $M$. Let $L^\bu$ be the simple
complex associated to $L^{p,q}=\Hom_R(K_q,M^p)$. There is a split
exact sequence
$$0\longrightarrow\Ext\big(H_{q-1}(K_\bu),M\big)\longrightarrow H^q(L^\bu)\longrightarrow
\Hom\big(H_q(K_\bu),M\big)\longrightarrow 0.$$
\endclaim

\proof{Proof} As the functor $\Hom_R(\bu,M^p)$ is exact, we get
$$\eqalign{
{}_LE^{p,q}_1&=\Hom\big(H_q(K_\bu),M^p\big),\cr
{}_LE^{p,q}_2&=\cases{
\Hom\big(H_q(K_\bu),M\big)&for~~$p=0$,\cr
\Ext\big(H_q(K_\bu),M\big)&for~~$p=1$,\cr
0&for~~$p\ge 2$.\cr}\cr}$$
The spectral sequence collapses in $E_2$, therefore we get
$$\eqalign{
G_0\big(H^q(L^\bu)\big)&=\Hom\big(H_q(K_\bu),M\big),\cr
G_1\big(H^q(L^\bu)\big)&=\Ext\big(H_{q-1}(K_\bu),M\big)\cr}$$
and the expected exact sequence follows. By the same arguments as at 
the end of the proof of Formula~15.5, we may assume that $K_\bu$ is 
split, so that there is a projection $K_q\longrightarrow Z_q$. Then the composite
morphism
$$\eqalign{
\Hom\big(H_q(K_\bu),M\big)=\Hom(Z_q/B_q,M)\longrightarrow
\Hom&(K_q/B_q,M)\cr
&\subset Z^q(L^\bu)\longrightarrow H^q(L^\bu)\cr}$$
defines a splitting of the exact sequence.\qed
\endproof

\subsection{16.B}{Poincar\'e Duality for Sheaves}
Let $\cA$ be a sheaf of abelian groups on a locally compact topological
space $X$ of finite topological dimension $n=\topdim X$. By 13.12~c),
$\cA$ admits a soft resolution $\cL^\bu$ of length $n$. If
$M\to M^0\to M^1\to 0$ is an injective resolution of a $R$-module $M$,
we introduce the double complex of presheaves $\cF^{p,q}_{\cA,M}$
defined by
$$\cF_{\cA,M}^{p,q}(U)=\Hom_R\big(\cL^{n-q}_c(U),M^p\big),\leqno(16.6)$$
where the restriction map $\cF_{\cA,M}^{p,q}(U)\longrightarrow\cF_{\cA,M}^{p,q}(V)$ is the
adjoint of the inclusion map $\cL^{n-q}_c(V)\longrightarrow\cL^{n-q}_c(U)$ 
when $V\subset U$. As $\cL^{n-q}$ is soft, any $f\in\cL^{n-q}_c(U)$
can be written as $f=\sum f_\alpha$ with $(f_\alpha)$ subordinate
to any open covering $(U_\alpha)$ of $U$~; it follows easily that
$\cF_{\cA,M}^{p,q}$ satisfy axioms (II-2.4) of sheaves. The injectivity of $M^p$
implies that $\cF_{\cA,M}^{p,q}$ is a flabby sheaf. By Lemma~16.5, we get
a split exact sequence
$$\leqalignno{
0\longrightarrow\Ext\big(H^{n-q+1}_c(X,\cA),M\big)\longrightarrow
H^q&\big(\cF_{\cA,M}^\bu(X)\big)\cr
&\longrightarrow\Hom\big(H^{n-q}_c(X,\cA),M\big)\longrightarrow 0.&(16.7)\cr}$$
This can be seen as an abstract Poincar\'e duality formula, relating
the cohomology groups of a differential sheaf $\cF^\bu_{\cA,M}$
``dual" of $\cA$ to the dual of the cohomology with compact support
of $\cA$. In concrete applications, it still remains to compute
$H^q\big(\cF^\bu_{\cA,M}(X)\big)$. This can be done easily when $X$
is a manifold and $\cA$ is a constant or locally constant sheaf.

\subsection{16.C}{Poincar\'e Duality on Topological Manifolds}
Here, $X$ denotes a paracompact topological manifold of dimension $n$.

\claim{(16.8) Definition} Let $L$ be a $R$-module. A locally constant 
sheaf of stalk $L$ on $X$ is a sheaf $\cA$ such that every point has a
neighborhood $\Omega$ on which $\cA_{\restriction\Omega}$ is
$R$-isomorphic to the constant sheaf $L$.
\endclaim

Thus, a locally constant sheaf $\cA$ can be seen as a discrete fiber bundle
over $X$ whose fibers are $R$-modules and whose transition automorphisms are
$R$-linear. If $X$ is locally contractible, a locally constant sheaf 
of stalk $L$ is given, up to isomorphism, by a representation 
$\rho:\pi_1(X)\longrightarrow\Aut_R(L)$ of the fundamental group of $X$,
up to conjugation; denoting by $\smash{\wt X}$ the universal covering 
of $X$, the sheaf $\cA$ associated to $\rho$ can be viewed as the 
quotient of $\smash{\wt X}\times L$ by the diagonal action of $\pi_1(X)$.
We leave the reader check himself the details of these assertions:
in fact similar arguments will be explained in full details in \S V-6
when properties of flat vector bundles are discussed.

Let $\cA$ be a locally constant sheaf of stalk $L$, let
$\cL^\bu$ be a soft resolution of $\cA$ and $\cF^{p,q}_{\cA,M}$
the associated flabby sheaves. For an arbitrary open set \hbox{$U\subset X$},
Formula (16.7) gives a (non canonical) isomorphism
$$H^q\big(\cF_{\cA,M}^\bu(U)\big)\simeq\Hom\big(H^{n-q}_c(U,\cA),M\big)\oplus
\Ext\big(H^{n-q+1}_c(U,\cA),M\big)$$
and in the special case $q=0$ a canonical isomorphism
$$H^0\big(\cF_{\cA,M}^\bu(U)\big)=\Hom\big(H^n_c(U,\cA),M\big).\leqno(16.9)$$
For an open subset $\Omega$ homeomorphic to $\bR^n$, we have
$\cA_{\restriction\Omega}\simeq L$. Proposition 14.16 and the
exact sequence of the pair yield
$$H^q_c(\Omega,L)\simeq H^q(S^n,\{\infty\}\,;\,L)=
\cases{L&for~~$q=n$,\cr  0&for~~$q\ne n$.\cr}$$
If $\Omega\simeq\bR^n$, we find 
$$H^0\big(\cF_{\cA,M}^\bu(\Omega)\big)\simeq\Hom(L,M),~~~~
H^1\big(\cF_{\cA,M}^\bu(\Omega)\big)\simeq\Ext(L,M)$$
and $H^q\big(\cF_{\cA,M}^\bu(\Omega)\big)=0$ for $q\ne 0,1$.
Consider open
sets $V\subset\Omega$ where $V$ is a deformation retract of $\Omega$.
Then the restriction map $H^q\big(\cF_{\cA,M}^\bu(\Omega)\big)\longrightarrow
H^q\big(\cF_{\cA,M}^\bu(V)\big)$ is an isomorphism. Taking the direct 
limit over all such neighborhoods $V$ of a given point $x\in\Omega$, 
we see that $\cH^0(\cF_{\cA,M}^\bu)$ and $\cH^1(\cF_{\cA,M}^\bu)$ are
locally constant sheaves of stalks $\Hom(L,M)$ and $\Ext(L,M)$,
and that $\cH^q(\cF_{\cA,M}^\bu)=0$ for $q\ne 0,1$.
When $\Ext(L,M)=0$, the complex $\cF^\bu_{\cA,M}$ is thus a
flabby resolution of $\cH^0=\cH^0(\cF^\bu_{\cA,M})$ and we get
isomorphisms
$$\leqalignno{
&H^q\big(\cF^\bu_{\cA,M}(X)\big)=H^q(X,\cH^0),&(16.10)\cr
&\cH^0(U)=H^0(\cF^\bu_{\cA,M}(U)\big)=\Hom\big(H^n_c(U,\cA),M\big).
&(16.11)\cr}$$

\claim{(16.12) Definition} The locally constant sheaf $\tau_X=
\cH^0(\cF^\bu_{\bZ,\bZ})$ of stalk $\bZ$ defined by
$$\tau_X(U)=\Hom_\bZ\big(H^n_c(U,\bZ),\bZ\big)$$
is called the orientation sheaf (or dualizing sheaf) of $X$.
\endclaim

This sheaf is given by a homomorphism $\pi_1(X)\longrightarrow\{1,-1\}$~; it is
not difficult to check that $\tau_X$ coincides with the trivial sheaf
$\bZ$ if and only if $X$ is orientable (cf.\ exercice 18.?). In general,
$H^n_c(U,\cA)=H^n_c(U,\bZ)\otimes_\bZ\cA(U)$ for any small open
set $U$ on which $\cA$ is trivial, thus
$$\cH^0(\cF^\bu_{\cA,M})=\tau_X\otimes_\bZ\Hom(\cA,M).$$
A combination of (16.7) and $(16.10)$ then gives:

\claim{(16.13) Poincar\'e duality theorem} Let $X$ be a topological
manifold, let $\cA$ be a locally constant sheaf over $X$ of stalk $L$ and
let $M$ be a $R$-module such that $\Ext(L,M)=0$. There is a 
split exact sequence
$$\eqalign{
0\longrightarrow\Ext\big(H^{n-q+1}_c(X,\cA),M\big)\longrightarrow
H^q&\big(X,\tau_X\otimes\Hom(\cA,M)\big)\cr
&\longrightarrow\Hom\big(H^{n-q}_c(X,\cA),M\big)\longrightarrow 0.\cr}$$
In particular, if either $X$ is orientable or $R$ has characteristic 2,
then
$$\eqalignno{
0\longrightarrow\Ext\big(H^{n-q+1}_c(X,R),R\big)\longrightarrow H^q(X,R)\longrightarrow
\Hom\big(H^{n-q}_c&(X,R),R\big)\cr
&\longrightarrow 0.&\square\cr}$$
\endclaim

\claim{(16.14) Corollary} Let $X$ be a connected topological manifold,
$n=\dim X$. Then for any $R$-module $L$
\medskip
\item{\rm a)} $H^n_c(X,\tau_X\otimes L)\simeq L~;$
\medskip
\item{\rm b)} $H^n_c(X,L)\simeq L/2L$~~if $X$ is not orientable.
\endclaim

\proof{Proof} First assume that $L$ is free. For $q=0$ and $\cA=\tau_X\otimes L$,
the Poincar\'e duality formula gives an isomorphism
$$\Hom\big(H^n_c(X,\tau_X\otimes L),M\big)\simeq\Hom(L,M)$$
and the isomorphism is functorial with respect to morphisms $M\longrightarrow M'$.
Taking $M=L$ or $M=H^n_c(X,\tau_X\otimes L)$, we easily obtain the
existence of inverse morphisms 
$$
H^n_c(X,\tau_X\otimes L)\longrightarrow L\quad\hbox{and}\quad
L\longrightarrow H^n_c(X,\tau_X\otimes L),
$$
hence equality~a). Similarly, for $\cA=L$ we get
$$\Hom\big(H^n_c(X,L),M\big)\simeq H^0\big(X,\tau_X\otimes\Hom(L,M)\big).$$
If $X$ is non orientable, then $\tau_X$ is non trivial and the global
sections of the sheaf $\tau_X\otimes\Hom(L,M)$ consist of $2$-torsion
elements of $\Hom(L,M)$, that is
$$\Hom\big(H^n_c(X,L),M\big)\simeq\Hom(L/2L,M).$$
Formula b) follows. If $L$ is not free, the result can be extended by
using a free resolution $0\to L_1\to L_0\to L\to 0$ and the associated
long exact sequence.\qed
\endproof

\claim{(16.15) Remark} \rm If $X$ is a connected non compact
$n$-dimensional manifold, it can be proved (exercise 18.?) that
$H^n(X,\cA)=0$ for every locally constant sheaf $\cA$ on $X$.\qed
\endclaim

Assume from now on that $X$ is oriented. Replacing $M$ by $L\otimes M$
and using the obvious morphism $M\longrightarrow\Hom(L,L\otimes M)$, the Poincar\'e
duality theorem yields a morphism
$$H^q(X,M)\longrightarrow\Hom\big(H^{n-q}_c(X,L),L\otimes M\big),\leqno(16.16)$$
in other words, a bilinear pairing
$$H^{n-q}_c(X,L)\times H^q(X,M)\longrightarrow L\otimes M.\leqno(16.16')$$

\claim{(16.17) Proposition} Up to the sign, the above pairing is 
given by the cup product, modulo the identification
$H^n_c(X,L\otimes M)\simeq L\otimes M$.
\endclaim

\proof{Proof} By functoriality in $L$, we may assume $L=R$. Then we make the
following special choices of resolutions:
$$\eqalign{
\cL^q&=R^{[q]}~~~\hbox{\rm for}~~q<n,~~~~
       \cL^n=\ker(R^{[q]}\longrightarrow R^{[q+1]}),\cr
M^0&=\hbox{an injective module containing}~~
       M^{[n]}_c(X)/d^{n-1}M^{[n-1]}_c(X).\cr}$$
We embed $M$ in $M^0$ by $\lambda\mapsto u\otimes_\bZ\lambda$ where 
$u\in\bZ^{[n]}(X)$ is a representative of a generator of $H^n_c(X,\bZ)$,
and we set $M^1=M^0/M$. The projection map $M^0\longrightarrow M^1$ can be seen
as an extension of
$$\wt d^n~:~~M^{[n]}_c(X)/d^{n-1}M^{[n-1]}_c(X)\longrightarrow d^n M^{[n]}_c(X),$$
since Ker$\,\wt d^n\simeq H^n_c(X,M)=M$. The inclusion
$d^n M^{[n]}_c(X)\subset M^1$ can be extended into a map 
$\pi:M^{[n+1]}_c(X)\longrightarrow M^1$. The cup product bilinear map
$$M^{[q]}(U)\times R^{[n-q]}_c(U)\longrightarrow M^{[n]}_c(X)\longrightarrow M^0$$
gives rise to a morphism $M^{[q]}(U)\longrightarrow\cF^q_{R,M}(U)$ defined by
$$\cmalign{
&M^{[q]}(U)&\longrightarrow\Hom\big(\cL^{n-q}_c(U),M^0\big)&\oplus
              \Hom\big(\cL^{n-q+1}_c(U),M^1\big)\cr
&\hfill f&\longmapsto\hfill(g\longmapsto f\smallsmile g)~~~
&\oplus~~~\big(h\longmapsto\pi(f\smallsmile h)\big).\cr}\leqno(16.18)$$
This morphism is easily seen to give a morphism of differential sheaves 
$M^{[\bu]}\longrightarrow\cF^\bu_{R,M}$, when $M^{[\bu]}$ is truncated in degree $n$,
i.e.\ when $M^{[n]}$ is replaced by Ker$\,d^n$. The induced morphism
$$M=\cH^0(M^{[\bu]})\longrightarrow\cH^0(\cF^\bu_{R,M})$$
is then the identity map, hence the cup product morphism (16.18)
actually induces the Poincar\'e duality map (16.16).\qed
\endproof

\claim{(16.19) Remark} \rm If $X$ is an oriented differentiable manifold,
the natural isomorphism $H^n_c(X,\bR)\simeq\bR$ given by 16.14~a)
corresponds in De Rham cohomology to the integration morphism
$f\longmapsto\int_X f$, $f\in\cD_n(X)$. Indeed, by a partition of
unity, we may assume that $\Supp\,f\subset\Omega\simeq\bR^n$.
The proof is thus reduced to the case $X=\bR^n$, which itself
reduces to $X=\bR$ since the cup product is compatible with the wedge
product of forms. Let us consider the covering $\cU=(]k-1,k+1[)_{k\in\bZ}$
and a partition of unity $(\psi_k)$ subordinate to $\cU$. The \v Cech
differential
$$\eqalign{
AC^0(\cU,\bZ)&\longrightarrow AC^1(\cU,\bZ)\cr
(c_k)&\longmapsto (c_{k\,k+1})=(c_{k+1}-c_k)\cr}$$
shows immediately that the generators of $H^1_c(\bR,\bZ)$ are the
$1$-cocycles $c$ such that $c_{01}=\pm 1$ and $c_{k\,k+1}=0$ for
$k\ne 0$. By Formula (6.12), the associated closed differential
form is
$$f=c_{01}\psi_1d\psi_0+c_{10}\psi_0d\psi_1,$$
hence $f=\pm{\bf 1}_{[0,1]}d\psi_0$ and $f$ does satisfy
$\int_\bR f=\pm 1$.
\endclaim

\claim{(16.20) Corollary} If $X$ is an oriented $\cC^\infty$ manifold, the bilinear map
$$H^{n-q}_c(X,\bR)\times H^q(X,\bR)\longrightarrow\bR,~~~~
(\{f\},\{g\})\longmapsto\int_X f\wedge g$$
is well defined and identifies $H^q(X,\bR)$ to the dual of
$H^{n-q}_c(X,\bR)$.
\endclaim


\chapter{Chapter V}{Hermitian Vector Bundles}
\begineightpoint
This chapter introduces the basic definitions concerning vector bundles
and connections. In the first sections, the concepts of connection,
curvature form, first Chern class are considered in the framework of
differentiable manifolds. Although we are mainly interested in complex 
manifolds, the ideas which will be developed in the next chapters also
involve real analysis and real geometry as essential tools. In the
second part, the special features of connections over complex manifolds
are investigated in detail: Chern connections, first Chern class of
type $(1,1)$, induced curvature forms on sub- and  quotient bundles,
$\ldots\,$. These general concepts are then illustrated by the example
of universal vector bundles over $\bP^n$ and over Grassmannians.
\endeightpoint

\section{1}{Definition of Vector Bundles}
Let $M$ be a $\cC^\infty$ differentiable manifold of dimension $m$ and let
$\bK=\bR$ or $\bK=\bC$ be the scalar field. A (real, complex) {\it 
vector bundle} of rank $r$ over $M$ is a $\cC^\infty$ manifold $E$ together with
\medskip
\item{\rm i)} a $\cC^\infty$ map $\pi:E\longrightarrow M$ called the projection,
\smallskip
\item{\rm ii)} a $\bK$-vector space structure of dimension $r$ on each 
fiber $E_x=\pi^{-1}(x)$
\medskip
\noindent{}such that the vector space structure is {\it locally trivial}.
This means that there exists an open covering $(V_\alpha )_{\alpha \in I}$
of $M$ and $\cC^\infty$ diffeomorphisms called \hbox{\it trivializations}
$$\theta_\alpha:E_{\restriction V_\alpha}\longrightarrow V_\alpha\times\bK^r,~~~~
\hbox{\rm where}~~E_{\restriction V_\alpha}=\pi^{-1}(V_\alpha),$$
such that for every $x\in V_\alpha $ the map
$$E_x \buildo \theta_\alpha  \over \longrightarrow \{x\} \times \bK^r
\longrightarrow \bK^r$$
is a linear isomorphism. For each $\alpha ,\beta \in I$, the map
$$\theta_{\alpha \beta } = \theta_\alpha \circ \theta_\beta^{-1}:
(V_\alpha  \cap V_\beta ) \times \bK^r \longrightarrow (V_\alpha \cap V_\beta )
\times \bK^r$$
acts as a linear automorphism on each fiber $\{x\}\times \bK^r$. 
It can thus be written
$$\theta_{\alpha \beta }(x,\xi ) = (x,g_{\alpha \beta }(x)\cdot\xi ),~~~~
(x,\xi ) \in (V_\alpha \cap V_\beta )\times \bK^r$$
where $(g_{\alpha \beta })_{(\alpha ,\beta )\in I\times I}$ is a collection
of invertible matrices with coefficients in $\cC^\infty(V_\alpha \cap V_\beta,\bK)$,
satisfying the cocycle relation
$$g_{\alpha\beta}\,g_{\beta\gamma }=g_{\alpha\gamma}\quad\hbox{\rm on}
\quad V_\alpha\cap V_\beta\cap V_\gamma.\leqno(1.1)$$
The collection $(g_{\alpha \beta })$ is called a {\it system of transition
matrices}. Conversely, any collection of invertible matrices satisfying (1.1)
defines a vector bundle $E$, obtained by gluing the charts $V_\alpha\times
\bK^r$ via the identifications  $\theta_{\alpha\beta}$.

\claim{(1.2) Example} \rm The product manifold $E=M\times\bK^r$ is a vector
bundle over $M$, and is called the {\it trivial vector bundle} of rank $r$
over $M$. We shall often simply denote it $\bK^r$ for brevity.
\endclaim

\claim{(1.3) Example} \rm A much more interesting example of real vector bundle 
is the {\it tangent bundle} $TM$~; if $\tau_\alpha:V_\alpha\longrightarrow\bR^n$ 
is a collection of coordinate charts on $M$, then 
$\theta_\alpha=\pi\times d\tau_\alpha:TM_{\restriction V_\alpha}\longrightarrow 
V_\alpha\times\bR^m$ define trivializations of $TM$ 
and the transition matrices are given by $g_{\alpha\beta}(x)=
d\tau_{\alpha\beta}(x^\beta)$ where $\tau_{\alpha\beta}=
\tau_\alpha\circ\smash{\tau_\beta^{-1}}$ and $x^\beta=\tau_\beta(x)$.
The dual $T^\star M$ of $TM$ is called the {\it cotangent bundle} and 
the $p$-th exterior power $\Lambda^pT^\star M$ is called the bundle of 
differential forms of degree $p$ on $M$.
\endclaim

\claim{(1.4) Definition} If $\Omega\subset M$ is an open subset and 
$k$ a positive integer or $+\infty$, we let $\cC^k(\Omega ,E)$ denote
the space of $\cC^k$ sections of $\smash{E_{\restriction\Omega}}$, i.e.\ 
the space of $\cC^k$ maps 
$s: \Omega \longrightarrow E$ such that $s(x)\in E_x$ for all $x\in\Omega$
$($that is $\pi \circ s =\Id_\Omega)$.
\endclaim

Let $\theta:E_{\restriction V}\longrightarrow V\times\bK^r$ be a trivialization 
of $E$. To $\theta$, we associate the $\cC^\infty$ {\it frame}
$(e_1,\ldots,e_r)$ of $E_{\restriction V}$ defined by
$$e_\lambda(x)=\theta^{-1}(x,\varepsilon_\lambda),~~~~x\in V,$$
where $(\varepsilon_\lambda)$ is the standard basis of $\bK^r$.
A section $s\in \cC^k(V,E)$ can then be represented in terms of its 
components $\theta(s)=\sigma=(\sigma_1,\ldots,\sigma_r)$ by
$$s=\sum_{1\le\lambda\le r}\sigma_\lambda e_\lambda~~~\hbox{\rm on}~~V,
~~~~\sigma_\lambda\in \cC^k(V,\bK).$$
Let $(\theta_\alpha)$ be a family of trivializations relative to a covering
$(V_\alpha)$ of $M$. Given a global section $s\in \cC^k(M,E)$, the
components $\theta_\alpha(s)=\sigma^\alpha=
(\sigma^\alpha_1,\ldots,\sigma^\alpha_r)$ satisfy the {\it transition relations}
$$\sigma^\alpha=g_{\alpha\beta}\,\sigma^\beta~~~~\hbox{\rm on}~~
V_\alpha\cap V_\beta.\leqno(1.5)$$
Conversely, any collection of vector valued functions $\sigma^\alpha:
V_\alpha\longrightarrow\bK^r$ satisfying the transition relations defines a global 
section $s$ of $E$.

More generally, we shall also consider differential forms on $M$
with values in $E$. Such forms are nothing else than sections of the
tensor product bundle $\Lambda^pT^\star M\otimes_\bR E$. We shall write
$$\leqalignno{
\cC^k_p(\Omega ,E) &= \cC^k(\Omega ,\Lambda ^pT^{\star} M \otimes_\bR E)&(1.6)
\cr
\cC^k_\bullet(\Omega ,E) &= \bigoplus_{0\le p\le m}\cC^k_p(\Omega ,E).&(1.7)\cr}
$$

\section{2}{Linear Connections}
A (linear) connection $D$ on the bundle $E$ is a linear differential operator
of order 1 acting on $\cC^\infty_\bullet(M,E)$ and satisfying the following 
properties:
$$\leqalignno{
&D:\cC^\infty_q(M,E) \longrightarrow \cC^\infty_{q+1}(M,E),&(2.1)\cr&D(f\wedge s) = df\wedge s + (-1)^p f\wedge Ds&(2.1')\cr}$$
for any $f\in \cC^\infty_p(M,\bK)$ and $s\in \cC^\infty_q(M,E)$, where $df$ stands
for the usual exterior derivative of $f$.

Assume that $\theta:E_{\restriction\Omega} \to \Omega \times \bK^r$ is a trivialization of 
$E_{\restriction\Omega}$, and let  $(e_1,\ldots,e_r)$ be the corresponding frame
of $E_{\restriction\Omega}$. Then any $s\in \cC^\infty_q(\Omega ,E)$ can be written
in a unique way
$$s = \sum_{1\le \lambda \le r} \sigma_\lambda \otimes e_\lambda ,~~~~
\sigma_\lambda  \in \cC^\infty_q(\Omega ,\bK).$$
By axiom $(2.1')$ we get
$$Ds = \sum_{1\le \lambda \le r} \big(d\sigma_\lambda  \otimes e_\lambda  +
(-1)^p \sigma_\lambda  \wedge De_\lambda \big).$$
If we write $De_\mu  = \sum_{1\le \lambda \le r} a_{\lambda \mu } \otimes
e_\lambda $ where $a_{\lambda \mu } \in \cC^\infty_1(\Omega ,\bK)$, we thus have
$$Ds = \sum_\lambda  \big(d\sigma_\lambda  + \sum_\mu a_{\lambda \mu }\wedge 
\sigma_\mu \big) \otimes e_\lambda .$$
Identify $E_{\restriction\Omega}$ with $\Omega \times \bK^r$ via $\theta $
and denote by $d$ the trivial connection $d\sigma  = (d\sigma_\lambda )$ on
$\Omega \times \bK^r$. Then the operator $D$ can be written
$$Ds \simeq_\theta ~ d\sigma  + A\wedge \sigma \leqno(2.2)$$
where $A = (a_{\lambda \mu }) \in \cC^\infty_1(\Omega ,\Hom (\bK^r,\bK^r))$.
Conversely, it is clear that any operator $D$ defined in such a way is a
connection on $E_{\restriction\Omega}$. The matrix 1-form $A$ will be called the 
{\it connection form} of $D$ associated to the trivialization $\theta $.
If~$\wt \theta: E_{\restriction\Omega} \to \Omega \times \bK^r$ is another
trivialization and if we set
$$g=\wt\theta\circ\theta^{-1}\in \cC^\infty(\Omega,\Gl(\bK^r))$$
then the new components $\wt\sigma=(\wt\sigma_\lambda)$ are related to the old 
ones by $\wt\sigma=g\sigma$. Let~$\wt A$ be the connection form of $D$ with 
respect to $\wt\theta$. Then 
$$\eqalign{
Ds &\simeq_{\wt \theta } ~d\wt \sigma  + \wt A\wedge \wt 
\sigma\cr
Ds &\simeq_\theta ~ g^{-1}(d\wt\sigma + \wt A \wedge \wt \sigma )
= g^{-1}(d(g\sigma ) + \wt A \wedge g\sigma )\cr
   &=~~\,d\sigma  + (g^{-1}\wt Ag + g^{-1}dg)\wedge \sigma.\cr}$$
Therefore we obtain the {\it gauge transformation law\/}:
$$A = g^{-1}\wt Ag + g^{-1}dg.\leqno(2.3)$$

\section{3}{Curvature Tensor}
Let us compute  $D^2: \cC^\infty_q(M,E) \to \cC^\infty_{q+2}(M,E)$ with respect to the
trivialization $\theta:E_{\restriction\Omega} \to \Omega \times \bK^r$. 
We obtain
$$\eqalign{
D^2s &\simeq_\theta~d(d\sigma+A\wedge\sigma)+A\wedge(d\sigma+A\wedge
\sigma)\cr
&= d^2\sigma+(dA\wedge \sigma-A\wedge d\sigma)+(A\wedge d\sigma+A\wedge
A\wedge\sigma)\cr
&= (dA+ A\wedge A)\wedge \sigma.\cr}$$
It follows that there exists a global 2-form $\Theta(D) \in \cC^\infty_2(M,\Hom(E,E))$
called {\it the curvature tensor} of $D$, such that
$$D^2s = \Theta(D)\wedge s,$$
given with respect to any trivialization $\theta $ by
$$\Theta(D) \simeq_\theta ~ dA+A\wedge A.\leqno(3.1)$$

\claim{(3.2) Remark} \rm If $E$ is of rank $r=1$, then $A\in \cC^\infty_1(M,\bK)$
and $\Hom(E,E)$ is canonically isomorphic to the trivial bundle $M\times\bK$, 
because the endomorphisms of each fiber $E_x$ are homotheties. With the 
identification $\Hom(E,E)=\bK$, the curvature tensor $\Theta(D)$ can be considered
as a closed 2-form with values in $\bK$:
$$\Theta(D) = dA.\leqno(3.3)$$
In this case, the gauge transformation law can be written
$$A = \wt A + g^{-1}dg,~~~~
g=\wt\theta\circ\theta^{-1}\in \cC^\infty(\Omega ,\bK^\star).\leqno(3.4)$$
It is then immediately clear that $dA=d\wt A$, and this equality
shows again that $dA$ does not depend on $\theta$.\qed
\endclaim

Now, we show that the curvature tensor is closely related to
commutation properties of covariant derivatives.

\claim{(3.5) Definition} If $\xi$ is a $\cC^\infty$ vector field with values in
$TM$, the covariant derivative of a section $s\in \cC^\infty(M,E)$ in the
direction $\xi$ is the section \hbox{$\xi_D\cdot s\in \cC^\infty(M,E)$}
defined by $\xi_D\cdot s=Ds\cdot\xi$.
\endclaim

\claim{(3.6) Proposition} For all sections $s\in \cC^\infty(M,E)$ and 
all vector fields $\xi,\eta\in$\break $\cC^\infty(M,TM)$, we have
$$\xi_D\cdot(\eta_D\cdot s)-\eta_D\cdot(\xi_D\cdot s)=[\xi,\eta]_D\cdot s
+\Theta(D)(\xi,\eta)\cdot s$$
where $[\xi,\eta]\in \cC^\infty(M,TM)$ is the Lie bracket of $\xi,\eta$.
\endclaim

\proof{Proof} Let $(x_1,\ldots,x_m)$ be local coordinates on an open set
$\Omega\subset M$. Let $\theta:E_{\restriction\Omega}\longrightarrow\Omega\times\bK^r$
be a trivialization of $E$ and let $A$ be the
corresponding connection form. If $\xi=\sum\xi_j
\,\partial/\partial x_j$ and $A=\sum A_j\,dx_j$, we find
$$\xi_Ds\simeq_\theta(d\sigma+A\sigma)\cdot\xi=\sum_j\xi_j\Big(
{\partial\sigma\over\partial x_j}+A_j\cdot\sigma\Big).\leqno(3.7)$$
Now, we compute the above commutator $[\xi_D,\eta_D]$ at a given point
$z_0\in\Omega$. Without loss of generality, we may assume $A(z_0)=0$~;
in fact, one can always find a gauge transformation $g$ near $z_0$ such that
$g(z_0)=\Id$ and $dg(z_0)=A(z_0)$~; then (2.3) yields $\wt A(z_0)=0$.
If $\eta=\sum\eta_k\,\partial/\partial x_k$, we find $\eta_D\cdot s
\simeq_\theta\sum\eta_k\,\partial\sigma/\partial x_k$ at $z_0$, hence
$$\eqalign{
\eta_D\cdot(\xi_D\cdot s)&\simeq_\theta \sum_k\eta_k{\partial\over\partial x_k}
\sum_j\xi_j\Big({\partial\sigma\over\partial x_j}+A_j\cdot\sigma\Big),\cr
\xi_D\cdot(\eta_D\cdot s)&-\eta_D\cdot(\xi_D\cdot s)\simeq_\theta\cr
&\simeq_\theta\sum_{j,k}\Big(\xi_k{\partial\eta_j\over\partial x_k}-\eta_k
{\partial\xi_j\over\partial x_k}\Big){\partial\sigma\over\partial x_j}+
\sum_{j,k}{\partial A_j\over\partial x_k}(\xi_j\eta_k-\eta_j\xi_k)\cdot\sigma\cr
&=d\sigma([\xi,\eta])+dA(\xi,\eta)\cdot\sigma,\cr}$$
whereas $\Theta(D)\simeq_\theta dA$ and $[\xi,\eta]_Ds\simeq_\theta
d\sigma([\xi,\eta])$ at point $z_0$.\qed
\endproof

\section{4}{Operations on Vector Bundles}
Let  $E,F$ be vector bundles of rank $r_1,r_2$ over $M$.
Given any functorial operation on vector spaces, a 
corresponding operation can be defined on bundles by applying the operation on
each fiber. For example $E^\star$, $E\oplus F$, $\Hom (E,F)$ are defined by
$$(E^\star)_x = (E_x)^\star,~~~~(E\oplus F)_x=E_x\oplus F_x,~~~~
   \Hom(E,F)_x = \Hom(E_x,F_x).$$
The bundles $E$ and $F$ can be trivialized over the same covering
$V_\alpha$ of $M$ (otherwise take a common refinement). If 
$(g_{\alpha\beta})$ and $(\gamma_{\alpha\beta})$ are the transition
matrices of $E$ and $F$, then for example $E\otimes F$, $\Lambda^k E$, 
$E^\star$ are the bundles defined by the transition matrices
$g_{\alpha\beta}\otimes\gamma_{\alpha\beta}$, $\Lambda^k g_{\alpha\beta}$, 
$(g_{\alpha\beta}^\dagger)^{-1}$ where $\dagger$ denotes transposition.

Suppose now that $E,F$ are equipped with connections $D_E,D_F$.
Then natural connections can be associated to all derived bundles.
Let us mention a few cases. First, we let
$$D_{E\oplus F} = D_E \oplus D_F.\leqno(4.1)$$
It follows immediately that
$$\Theta(D_{E\oplus F}) = \Theta(D_E) \oplus \Theta(D_F).\leqno(4.1')$$
$D_{E\otimes F}$ will be defined in such a way that the usual formula for the
differentiation of a product remains valid. For every  $s\in \cC^\infty_\bullet(M,E)$,
$t\in \cC^\infty_\bullet(M,F)$, the wedge product $s\wedge t$ can be combined with the
bilinear map $E\times F\longrightarrow E\otimes F$ in order to obtain a section 
$s\wedge t\in \cC^\infty(M,E\otimes F)$ of degree $\deg\,s+\deg\,t$.
Then there exists a unique connection $D_{E\otimes F}$ such that
$$D_{E\otimes F}(s\wedge t) = D_Es\wedge t + (-1)^{\deg~s}s\wedge D_Ft.
\leqno(4.2)$$
As the products $s\wedge t$ generate $\cC^\infty_\bullet(M,E\otimes F)$, the uniqueness
is clear. If $E$, $F$ are trivial on an open set $\Omega\subset M$ and if
$A_E$, $A_F$, are their connection 1-forms, the induced connection 
$D_{E\otimes F}$ is given by the connection form $A_E\otimes\Id_F+
\Id_E\otimes A_F$. The existence follows. An easy computation shows that
$D^2_{E\otimes F}(s\wedge t)=D^2_Es\wedge t+s\wedge D^2_Ft$, thus
$$\Theta(D_{E\otimes F})=\Theta(D_E)\otimes\Id_F+\Id_E{}\otimes{}\Theta(D_F).\leqno(4.2')$$
Similarly, there are unique connections $D_{E^\star}$ and $D_{\Hom (E,F)}$ 
such that
$$\leqalignno{
(D_{E^\star}u)\cdot s&=d(u\cdot s)-(-1)^{\deg~u}u\cdot D_Es,&(4.3)\cr
(D_{\Hom (E,F)}v)\cdot s&=D_F(v\cdot s)-(-1)^{\deg~v}v\cdot D_Es&(4.4)\cr}$$
whenever  $s\in \cC^\infty_\bullet(M,E),~u\in \cC^\infty_\bullet(M,E^\star),~
v\in \cC^\infty_\bullet\big(\Hom(E,F)\big)$. It follows that
$$0=d^2(u\cdot s)=\big(\Theta(D_{E^\star})\cdot u\big)\cdot s+
u\cdot\big(\Theta(D_E)\cdot s\big).$$
If $\dagger$ denotes the transposition operator $\Hom (E,E)\to\Hom(E^{\star},
E^\star)$, we thus get
$$\Theta(D_{E^\star}) = -\Theta(D_E)^\dagger.\leqno(4.3')$$
With the identification $\Hom(E,F)=E^\star\otimes F$, Formula $(4.2')$
implies
$$\Theta(D_{\Hom(E,F)})=\Id_{E^\star}\otimes \Theta(D_F)-\Theta(D_E)^\dagger\otimes\Id_F.
\leqno(4.4')$$
Finally, $\Lambda^kE$ carries a natural connection $D_{\Lambda^kE}$. For 
every $s_1,\ldots,s_k$ in $\cC^\infty_\bullet (M,E)$ of respective degrees $p_1,\ldots,p_k$, 
this connection satisfies
$$\leqalignno{
&D_{\Lambda ^kE}(s_1\wedge \ldots \wedge s_k) = \sum_{1\le j\le k}
(-1)^{p_1+\ldots +p_{j{-}1}} s_1\wedge\ldots D_E s_j\ldots\wedge s_k,&(4.5)\cr
&\Theta(D_{\Lambda^kE})\cdot(s_1\wedge\ldots\wedge s_k)=
\sum_{1\le j\le k}s_1\wedge\ldots\wedge \Theta(D_E)\cdot s_j\wedge\ldots 
\wedge s_k.&(4.5')\cr}$$
In particular, the {\it determinant bundle}, defined by $\det E=\Lambda^rE$
where $r$ is the rank of $E$, has a curvature form given by
$$\Theta(D_{\det E}) =\cT_E\big(\Theta(D_E)\big)\leqno(4.6)$$
where $\cT_E: \Hom(E,E) \longrightarrow \bK$ is the trace operator.
As a conclusion of this paragraph, we mention the following simple
identity.

\claim{(4.7) Bianchi identity} $D_{\Hom(E,E)}\big(\Theta(D_E)\big)=0$.
\endclaim

\proof{Proof} By definition of $D_{\Hom(E,E)}$, we find for any
$s\in \cC^\infty(M,E)$
$$\eqalignno{
D_{\Hom(E,E)}\big(\Theta(D_E)\big)\cdot s
&= D_E\big(\Theta(D_E)\cdot s\big)-\Theta(D_E)\cdot(D_E s)\cr
&=D^3_E s - D^3_E s = 0.&\square\cr}$$
\endproof

\section{5}{Pull-Back of a Vector Bundle}
Let $\wt M$, $M$ be $\cC^\infty$ manifolds and $\psi:\wt M\to M$
a smooth map. If $E$ is a vector bundle on $M$, one can define in a natural
way a $\cC^\infty$ vector bundle $\wt\pi:\wt E\to\wt M$ and
a $\cC^\infty$ linear morphism $\Psi:\wt E\to E$ such that the diagram
$$\matrix{
\wt E &\buildo{\displaystyle\Psi}\over\longrightarrow&E\cr
\big\downarrow\rlap{$\displaystyle\wt\pi$}&&\big\downarrow
\rlap{$\displaystyle\pi$}\cr
\wt M&\buildo{\displaystyle\psi}\over \longrightarrow&M\cr}$$
commutes and such that $\Psi:\wt E_x\longrightarrow E_{\psi(x)}$ is an isomorphism
for every $x\in M$. The bundle $\wt E$ can be defined by
$$\wt E=\{(\wt x,\xi)\in\wt M\times E~;~\psi(\wt x)=\pi(\xi)\}\leqno(5.1)$$
and the maps $\wt\pi$ and $\Psi$ are then the restrictions to $\wt E$ of the
projections of $\wt M\times E$ on $\wt M$ and $E$ respectively.

If $\theta_\alpha:E_{\restriction V_\alpha}\longrightarrow V_\alpha\times\bK^r$ are
trivializations of $E$, the maps
$$\wt\theta_\alpha=\theta_\alpha\circ\Psi:
\wt E_{\restriction \psi^{-1}(V_\alpha)}\longrightarrow \psi^{-1}(V_\alpha)\times\bK^r$$
define trivializations of $\wt E$ with respect to the covering
$\wt V_\alpha=\psi^{-1}(V_\alpha)$ of $\wt M$. The corresponding system of
transition matrices is given by
$$\wt g_{\alpha\beta}=g_{\alpha\beta}\circ\psi~~~~\hbox{\rm on}~~
  \wt V_\alpha\cap\wt V_\beta.\leqno(5.2)$$

\claim{(5.3) Definition} $\wt E$ is termed the pull-back of $E$ under the map
$\psi$ and is denoted $\wt E=\psi^\star E$.
\endclaim

Let $D$ be a connection on $E$. If $(A_\alpha)$ is the collection of
connection forms of $D$ with respect to the $\theta_\alpha$'s, one can
define a connection $\wt D$ on $\wt E$ by the collection of connection
forms $\wt A_\alpha=\psi^\star A_\alpha\in \cC^\infty_1\big(\wt V_\alpha,
\Hom(\bK^r,\bK^r)\big)$, i.e.\ for every $\wt s\in \cC^\infty_p(\wt V_\alpha,\wt E)$
$$\wt D\wt s\simeq_{\wt\theta_\alpha} d\wt\sigma+\psi^\star A_\alpha\wedge
\wt\sigma.$$
Given any section $s\in \cC^\infty_p(M,E)$, one defines a pull back $\psi^\star s$
which is a section in $\cC^\infty_p(\wt M,\wt E)\,$: for $s=f\otimes u$,
$f\in \cC^\infty_p(M,\bK)$, $u\in \cC^\infty(M,E)$, set $\psi^\star s=\psi^\star f\otimes
(u\circ\psi)$. Then we have the formula
$$\wt D(\psi^\star s)=\psi^\star(Ds).\leqno(5.4)$$
Using (5.4), a simple computation yields
$$\Theta(\wt D)=\psi^\star(\Theta(D)).\leqno(5.5)$$

\section{6}{Parallel Translation and Flat Vector Bundles}
Let $\gamma:[0,1]\longrightarrow M$ be a smooth curve and $s:[0,1]\to E$ a $\cC^\infty$ 
section of $E$ along $\gamma$, i.e.\ a $\cC^\infty$ map $s$ such that
$s(t)\in E_{\gamma(t)}$ for all $t\in[0,1]$. Then $s$ can be viewed as
a section of $\smash{\wt E}=\gamma^\star E$ over $[0,1]$. The {\it covariant
derivative} of $s$ is the section of $E$ along $\gamma$ defined by
$${Ds\over dt}=\wt Ds(t)\cdot{d\over dt}\in E_{\gamma(t)},\leqno(6.1)$$
where $\wt D$ is the induced connection on $\wt E$. If $A$ is a connection
form of $D$ with respect to a trivialization $\theta:E_{\restriction\Omega}\longrightarrow\Omega\times\bK^r$,
we have $\wt Ds\simeq_\theta d\sigma+\gamma^\star A\cdot\sigma$, i.e.\
$${Ds\over dt}\simeq_\theta {d\sigma\over dt}+\big(A(\gamma(t))\cdot
\gamma'(t)\big)\cdot\sigma(t)~~~\hbox{\rm for}~~\gamma(t)\in\Omega.
\leqno(6.2)$$
For $v\in E_{\gamma(0)}$ given, the Cauchy uniqueness and existence theorem
for ordinary linear differential equations implies that there exists a unique
section $s$ of $\smash{\wt E}$ such that $s(0)=v$ and $Ds/dt=0$.

\claim{(6.3) Definition} The linear map
$$T_\gamma:E_{\gamma(0)}\longrightarrow E_{\gamma(1)},~~~~v=s(0)\longmapsto s(1)$$
is called parallel translation along $\gamma$.
\endclaim

If $\gamma=\gamma_2\gamma_1$ is the composite of two paths $\gamma_1$,
$\gamma_2$ such that $\gamma_2(0)=\gamma_1(1)$, it is clear
that $T_{\gamma}=T_{\gamma_2}\circ T_{\gamma_1}$, and the inverse path 
$\gamma^{-1}:t\mapsto\gamma(1-t)$ is such that
$T_{\gamma^{-1}}=T_\gamma^{-1}$. It follows that $T_\gamma$ is a linear
isomorphism from $E_{\gamma(0)}$ onto $E_{\gamma(1)}$.

More generally, if $h:W\longrightarrow M$ is a $\cC^\infty$ map from a domain
$W\subset\bR^p$ into $M$ and if $s$ is a section of $h^\star E$,
we define covariant derivatives $Ds/\partial t_j$, $1\le j\le p$, by
$\wt D=h^\star D$ and
$${Ds\over\partial t_j}=\wt Ds\cdot{\partial\over\partial t_j}.\leqno(6.4)$$
Since $\partial/\partial t_j$, $\partial/\partial t_k$ commute and since
$\Theta(\wt D)=h^\star\,\Theta(D)$, Prop.~3.6 implies
$${D\over\partial t_j}{Ds\over\partial t_k}-
{D\over\partial t_k}{Ds\over\partial t_j}=
\Theta(\wt D)\Big({\partial\over\partial t_j},{\partial\over\partial t_k}\Big)
\cdot s=\Theta(D)_{h(t)}\Big({\partial h\over\partial t_j},{\partial h\over
\partial t_k}\Big)\cdot s(t).\leqno(6.5)$$

\claim{(6.6) Definition} The connection $D$ is said to be flat if $\Theta(D)=0$.
\endclaim

Assume from now on that $D$ is flat. We then show that $T_\gamma$ only
depends on the homotopy class of $\gamma$. Let $h:[0,1]\times[0,1]\longrightarrow M$
be a smooth homotopy $h(t,u)=\gamma_u(t)$ from $\gamma_0$
to $\gamma_1$ with fixed end points $a=\gamma_u(0)$, $b=\gamma_u(1)$.
Let $v\in E_a$ be given and let $s(t,u)$ be such that $s(0,u)=v$ and
$Ds/\partial t=0$ for all $u\in[0,1]$. Then $s$ is $\cC^\infty$ in both
variables $(t,u)$ by standard theorems on the dependence of
parameters. Moreover (6.5) implies that the covariant derivatives 
$D/\partial t$, $D/\partial u$ commute. Therefore, if we set
$s'=Ds/\partial u$, we find $Ds'/\partial t=0$ with initial condition
$s'(0,u)=0$ (recall that $s(0,u)$ is a constant). The uniqueness of
solutions of differential equations implies that $s'$ is identically
zero on $[0,1]\times[0,1]$, in particular $T_{\gamma_u}(v)=s(1,u)$ must
be constant, as desired.

\claim{(6.7) Proposition} Assume that $D$ is flat.
If $\Omega$ is a simply connected open subset of
$M$, then $E_{\restriction\Omega}$ admits a $\cC^\infty$ parallel frame $(e_1,\ldots,e_r)$,
in the sense that $De_\lambda=0$ on $\Omega$, $1\le\lambda\le r$.
For any two simply connected open subsets $\Omega,\Omega'$
the transition automorphism between the corresponding parallel frames
$(e_\lambda)$ and $(e'_\lambda)$ is locally constant.
\endclaim

The converse statement ``$E$ {\it has parallel frames near every point
implies that} $\Theta(D)=0~$"
can be immediately verified from the equality $\Theta(D)=D^2$.

\proof{Proof} Choose a base point $a\in\Omega$ and define a linear
isomorphism $\Phi:\Omega\times E_a\longrightarrow E_{\restriction\Omega}$ by sending $(x,v)$ on 
$T_\gamma(v)\in E_x$, where $\gamma$ is any path from $a$ to $x$ in $\Omega$
(two such paths are always homotopic by hypothesis). Now, for any path
$\gamma$ from $a$ to $x$, we have by construction $(D/dt)\Phi(\gamma(t),v)
=0$. Set $e_v(x)=\Phi(x,v)$. As $\gamma$ may reach any point 
$x\in\Omega$ with an arbitrary tangent vector $\xi=\gamma'(1)\in T_xM$, 
we get $De_v(x)\cdot\xi=(D/dt)\Phi(\gamma(t),v)_{\restriction t=1}=0$. Hence
$De_v$ is parallel for any fixed vector $v\in E_a$~; Prop.~6.7
follows.\qed
\endproof

Assume that $M$ is connected. Let $a$ be a base point and $\wt M\longrightarrow M$ the 
universal covering of $M$. The manifold $\wt M$ can be considered as the 
set of pairs $(x,[\gamma])$, where $[\gamma]$ is a homotopy class of 
paths from $a$ to $x$.
Let $\pi_1(M)$ be the fundamental group of $M$ with base point $a$, acting on
$\wt M$ on the left by $[\kappa]\cdot(x,[\gamma])=(x,[\gamma\kappa^{-1}])$.
If $D$ is flat, $\pi_1(M)$ acts also on $E_a$ by $([\kappa],v)
\mapsto T_\kappa(v)$, $[\kappa]\in\pi_1(M)$, $v\in E_a$, and we have a well
defined map
$$\Psi:\wt M\times E_a\longrightarrow E,~~~~\Psi(x,[\gamma])=T_\gamma(v).$$
Then $\Psi$ is invariant under the left action of $\pi_1(M)$ on 
$\wt M\times E_a$ defined by
$$[\kappa]\cdot\big((x,[\gamma]),v\big)=\big((x,[\gamma\kappa^{-1}]),
T_\kappa(v)\big),$$
therefore we have an isomorphism $E\simeq(\wt M\times E_a)/\pi_1(M)$.

Conversely, let $S$ be a $\bK$-vector space of dimension $r$ together
with a left action of $\pi_1(M)$. The quotient $E=(\wt M\times S)/\pi_1(M)$ is
a vector bundle over $M$ with locally constant transition automorphisms
$(g_{\alpha\beta})$ relatively to any covering $(V_\alpha)$ of $M$ by simply
connected open sets. The relation $\sigma^\alpha=g_{\alpha\beta}\,
\sigma^\beta$ implies $d\sigma^\alpha=g_{\alpha\beta}\,d\sigma^\beta$
on $V_\alpha\cap V_\beta$. We may therefore define a connection $D$ on $E$
by letting $Ds\simeq_{\theta_\alpha}d\sigma^\alpha$ on each $V_\alpha$.
Then clearly $\Theta(D)=0$.

\section{7}{Hermitian Vector Bundles and Connections}
A complex vector bundle $E$ is said to be {\it hermitian} if a positive definite
hermi\-tian form $|~~|^2$ is given on each fiber $E_x$ in such a way that the
map $E\to \bR_+,~\xi \mapsto |\xi |^2$ is smooth. The notion
of a euclidean (real) vector bundle is similar, so we leave the reader
adapt our notations to that case.

Let $\theta:E_{\restriction\Omega}\longrightarrow\Omega\times\bC^r$ be a 
trivialization and let $(e_1,\ldots,e_r)$ be the corres\-ponding
frame of $E_{\restriction\Omega}$. The associated inner product of $E$ 
is given by a positive definite hermitian matrix $(h_{\lambda\mu})$ with 
$\cC^\infty$ coefficients on $\Omega$, such that
$$\langle e_\lambda(x),e_\mu(x)\rangle=h_{\lambda\mu}(x),~~~~
\forall x\in\Omega.$$
When $E$ is hermitian, one can define a natural sesquilinear map
$$\leqalignno{
\cC^\infty_p(M,E) \times \cC^\infty_q(M,E) &\longrightarrow \cC^\infty_{p+q}(M,\bC)\cr
(s,t) &\longmapsto \{ s,t\}&(7.1)\cr}$$
combining the wedge product of forms with the hermitian metric on $E$~;
\hbox{if $s=\sum\sigma_\lambda\otimes e_\lambda$,}
$t=\sum\tau_\mu\otimes e_\mu$, we let
$$\{ s,t\}=\sum_{1\le\lambda,\mu\le r}\sigma_\lambda\wedge\ol\tau_\mu
\,\langle e_\lambda,e_\mu\rangle.$$
A connection $D$ is said to be compatible with the hermitian structure of
$E$, or briefly {\it hermitian}, if for every $s\in \cC^\infty_p(M,E),~t\in
\cC^\infty_q(M,E)$ we have
$$d\{ s,t\} = \{ Ds,t\} + (-1)^p \{ s,Dt\}.
\leqno(7.2)$$
Let $(e_1,\ldots,e_r)$ be an {\it orthonormal frame} of $E_{\restriction\Omega}$.
Denote $\theta(s)=\sigma=(\sigma_\lambda)$ and 
$\theta(t)=\tau=(\tau_\lambda)$. Then
$$\eqalign{
\{s,t\}&=\{\sigma,\tau\}=\sum_{1\le\lambda\le r}\sigma_\lambda\wedge\ol
\tau_\lambda,\cr
d\{ s,t\}&= \{ d\sigma ,\tau \} + (-1)^p \{ 
\sigma ,d\tau\}.\cr}$$
Therefore $D_{\restriction\Omega}$ is hermitian if and only if its connection 
form $A$ satisfies
$$\{A\sigma,\tau \} + (-1)^p \{\sigma ,A\tau\}
= \{(A+A^\star)\wedge \sigma ,\tau \}=0$$
for all $\sigma ,\tau $, i.e.\
$$A^\star=-A\quad\hbox{\rm or}\quad(\ol{a_{\mu\lambda}})=-(a_{\lambda\mu}).
\leqno(7.3)$$
This means that $\ii A$ is a 1-form with values in the space $\Herm
(\bC^r,\bC^r)$ of hermi\-tian matrices. The identity $d^2\{s,t\}=0$
implies $\{D^2s,t\}+\{s,D^2t\}=0$, i.e.\
$\{\Theta(D)\wedge s,t\}+\{s,\Theta(D)\wedge t\}=0$.
Therefore $\Theta(D)^\star=-\Theta(D)$ and the curvature tensor $\Theta(D)$ is such that
$$\ii\,\Theta(D)\in \cC^\infty_2(M,\Herm(E,E)).$$

\claim{(7.4) Special case} \rm If $E$ is a hermitian line bundle $(r=1)$, 
$\smash{D_{\restriction\Omega}}$ is a hermitian 
connection if and only if its connection form $A$ associated to any given 
orthonormal frame of $E_{\restriction\Omega}$ is a 1-form with purely imaginary 
values.
\endclaim

If $\theta ,\wt \theta:\smash{E_{\restriction\Omega}}\to\Omega$ are two such 
trivializations on a simply connected open subset
$\Omega \subset M$, then $g=\smash{\wt\theta}\circ\theta^{-1}=e^{\ii\varphi}$ 
for some real phase function $\varphi\in \cC^\infty(\Omega,\bR)$. The gauge
transformation law can be written
$$A=\wt A+\ii\,d\varphi .$$
In this case, we see that $\ii\,\Theta(D) \in \cC^\infty_2(M,\bR).$

\claim{(7.5) Remark} \rm If $s,s'\in \cC^\infty(M,E)$ are two sections of $E$ along a 
smooth curve $\gamma:[0,1]\longrightarrow M$, one can easily verify the formula
$${d\over dt}\langle s(t),s'(t)\rangle=
\langle{Ds\over dt},s'\rangle+\langle s,{Ds'\over dt}\rangle.$$
In particular, if~ $(e_1,\ldots,e_r)$~ is a parallel frame of $E$ along $\gamma$
such that $\big(e_1(0),\ldots,e_r(0)\big)$ is orthonormal, then
$\big(e_1(t),\ldots,e_r(t)\big)$ is orthonormal for all $t$. All parallel 
translation operators $T_\gamma$ defined in \S 6 are thus isometries 
of the fibers. It follows that $E$ has a flat hermitian connection $D$
if and only if $E$ can be defined
by means of locally constant unitary transition automorphisms 
$g_{\alpha\beta}$, or equivalently if $E$
is isomorphic to the hermitian bundle $(\wt M\times S)/\pi_1(M)$
defined by a unitary representation of $\pi_1(M)$ in a hermitian
vector space $S$. Such a bundle $E$ is said to be {\it hermitian flat}.
\endclaim

\section{8}{Vector Bundles and Locally Free Sheaves}
We denote here by $\cE$ the sheaf of germs of $\cC^\infty$ complex functions on
$M$. Let $F\longrightarrow M$ be a $\cC^\infty$ complex vector bundle of rank $r$. We let
$\cF$ be the sheaf of germs of $\cC^\infty$ sections of $F$, i.e.\ the sheaf 
whose space of sections on an open subset \hbox{$U\subset M$} is $\cF(U)
=\cC^\infty(U,F)$. It is clear that $\cF$ is a $\cE$-module. Furthermore,
if $F_{\restriction\Omega}\simeq\Omega\times\bC^r$ is trivial, the sheaf 
$\cF_{\restriction\Omega}$ is isomorphic to 
$\cE^r_{\restriction\Omega}$ as a 
$\cE_{\restriction\Omega}$-module.

\claim{(8.1) Definition} A sheaf $\cS$ of modules over a sheaf of rings
$\cR$ is said to be locally free of rank $k$ if every point in the base
has a neighborhood $\Omega$ such that $\cS_{\restriction\Omega}$ is
$\cR$-isomorphic to $\cR^k_{\restriction\Omega}$.
\endclaim

Suppose that $\cS$ is a locally free $\cE$-module of rank $r$.
There exists a covering
$(V_\alpha)_{\alpha\in I}$ of $M$ and sheaf isomorphisms
$$\theta_\alpha:\cS_{\restriction V_\alpha}\longrightarrow \cE^r_{\restriction 
V_\alpha}.$$
Then we have transition isomorphisms $g_{\alpha\beta}=\theta_\alpha\circ
\theta_\beta^{-1}:\cE^r\to\cE^r$ defined on $V_\alpha\cap V_\beta$,
and such an isomorphism is the multiplication by an invertible matrix with 
$\cC^\infty$ coefficients on $V_\alpha\cap V_\beta$. The concepts of
vector bundle and of locally free $\cE$-module are thus completely 
equivalent.

Assume now that $F\longrightarrow M$ is a line bundle ($r=1$). Then every collection of
transition automorphisms $g=(g_{\alpha\beta})$ defines a \v Cech 1-cocycle
with values in the multiplicative sheaf $\cE^\star$ of invertible 
$\cC^\infty$ functions on $M$. In fact the definition of the \v Cech differential 
(cf.\ (IV-5.1)) gives $(\delta g)_{\alpha\beta\gamma}=g_{\beta\gamma}
g_{\alpha\gamma}^{-1}g_{\alpha\beta}$, and we have $\delta g=1$ in view of 
(1.1). Let $\theta'_\alpha$ be another family of trivializations
and $(g'_{\alpha\beta})$ the associated cocycle (it is no loss of generality
to assume that both are defined on the same covering since we may otherwise
take a refinement). Then we have 
$$\theta'_\alpha\circ\theta_\alpha^{-1}:
V_\alpha\times\bC\longrightarrow V_\alpha\times\bC,~~~~(x,\xi)\longmapsto 
(x,u_\alpha(x)\xi),~~~~u_\alpha\in\cE^\star(V_\alpha).$$
It follows that $g_{\alpha\beta}=g'_{\alpha\beta}u_\alpha^{-1}u_\beta$, i.e.\
that the \v Cech 1-cocycles $g,g'$ differ only by the \v Cech 1-coboundary 
$\delta u$.
Therefore, there is a well defined map which associates to every line bundle 
$F$ over $M$ the \v Cech cohomology class $\{g\}\in H^1(M,\cE^\star)$
of its cocycle of transition automorphisms. It is easy to verify that the
cohomology classes associated to two line bundles $F,F'$ are equal if and only
if these bundles are isomorphic: if~ $g=g'\cdot\delta u$, then the collection
of maps
$$\eqalign{
F_{\restriction V_\alpha}\buildo\theta_\alpha\over\longrightarrow V_\alpha\times\bC&\longrightarrow
V_\alpha\times\bC\buildo\theta_\alpha^{\prime -1}\over\longrightarrow F'_{\restriction 
V_\alpha}\cr(x,\xi)&\longmapsto(x,u_\alpha(x)\xi)\cr}$$
defines a global isomorphism $F\to F'$. It is clear that the multiplicative
group structure on $H^1(M,\cE^\star)$ corresponds to the tensor product
of line bundles (the inverse of a line bundle being given by its dual). We may
summarize this discussion by the following:

\claim{(8.2) Theorem} The group of isomorphism classes of complex $\cC^\infty$ line
bundles is in one-to-one correspondence with the \v Cech
cohomology group $H^1(M,\cE^\star)$.
\endclaim

\section{9}{First Chern Class}
Throughout this section, we assume that $E$ is a complex line bundle 
(that is, rk$\,E=r=1$).
Let D be a connection on $E$. By (3.3), $\Theta(D)$ is a closed 2-form
on $M$. Moreover, if $D'$ is another connection on $E$, then (2.2) shows
that $D'=D+\Gamma\wedge\bu$~ where $\Gamma\in \cC^\infty_1(M,\bC)$. By (3.3), we get
$$\Theta(D')=\Theta(D)+d\Gamma.\leqno(9.1)$$
This formula shows that the De Rham class $\{\Theta(D)\}\in
H^2_{DR}(M,\bC)$ does not depend on the particular choice of $D$. If $D$ is
chosen to be hermitian with respect to a given hermitian metric on $E$
(such a connection can always be constructed by means of a partition of unity) 
then $\ii\,\Theta(D)$ is a real 2-form, thus $\{\ii\,\Theta(D)\}\in H^2_{DR}(M,\bR)$.
Consider now the one-to-one correspondence given by Th.~8.2:
$$\eqalign{
\{\hbox{\rm isomorphism classes of line bundles}\}&\longrightarrow H^1(M,\cE^\star)\cr
\hbox{\rm class}~\{E\}~\hbox{\rm defined by the cocycle}~(g_{\alpha\beta})
&\longmapsto\hbox{\rm class of}~(g_{\alpha\beta}).\cr}$$
Using the exponential exact sequence of sheaves
$$\eqalign{0\longrightarrow\bZ\longrightarrow\cE&\longrightarrow\cE^\star\longrightarrow 1\cr
f&\longmapsto e^{2\pi\ii f}\cr}$$
and the fact that $H^1(M,\cE)=H^2(M,\cE)=0$, we obtain:

\claim{(9.2) Theorem and Definition} The coboundary morphism
$$H^1(M,\cE^\star)\buildo\partial\over\longrightarrow H^2(M,\bZ)$$
is an isomorphism. The first Chern class of a line bundle $E$ is
the image $c_1(E)$ in $H^2(M,\bZ)$ of the \v Cech cohomology class of the 
1-cocycle $(g_{\alpha\beta})$ asso\-ciated to $E\,$:
$$c_1(E)=\partial\{(g_{\alpha\beta})\}.\leqno(9.3)$$
\endclaim

Consider the natural morphism
$$H^2(M,\bZ)\longrightarrow H^2(M,\bR)\simeq H^2_{DR}(M,\bR)\leqno(9.4)$$
where the isomorphism $\simeq$ is that given by the De Rham-Weil
isomorphism theorem and the sign convention of Formula (IV-6.11).

\claim{(9.5) Theorem} The image of $c_1(E)$ in $H^2_{DR}(M,\bR)$ under 
{\rm (9.4)} coincides with the De Rham cohomology class
$\{{\ii\over 2\pi}\Theta(D)\}$ associated to any (hermitian) connection $D$ on $E$.
\endclaim

\proof{Proof} Choose an open covering $(V_\alpha)_{\alpha\in I}$ of $M$
such that $E$ is trivial on each $V_\alpha$, and such that all intersections
$V_\alpha\cap V_\beta$ are simply connected (as in \S IV-6, choose the 
$V_\alpha$ to be small balls relative to a given locally finite covering of 
$M$ by coordinate patches). Denote by $A_\alpha$ the
connection forms of $D$ with respect to a family of isometric trivializations
$$\theta_\alpha:~E_{\restriction V_\alpha}\longrightarrow V_\alpha\times\bC^r.$$
Let $g_{\alpha\beta}\in\cE^\star(V_\alpha\cap V_\beta)$ be the 
corresponding transition automorphisms. Then $|g_{\alpha\beta}|=1$, and
as $V_\alpha\cap V_\beta$ is simply connected, we may choose real functions 
$u_{\alpha\beta}\in\cE(V_\alpha\cap V_\beta)$ such that
$$g_{\alpha\beta}=\exp(2\pi\ii\,u_{\alpha\beta}).$$
By definition, the first Chern class $c_1(E)$ is the \v Cech 2-cocycle 
$$\eqalign{
c_1(E)=&\partial\{(g_{\alpha\beta})\}=\{(\delta u)_{\alpha\beta\gamma})\}
\in H^2(M,\bZ)~~~~\hbox{\rm where}\cr
(\delta u)_{\alpha\beta\gamma}:=&u_{\beta\gamma}-u_{\alpha\gamma}+
u_{\alpha\beta}.\cr}$$
Now, if $\cE^q$ (resp.\ $\cZ^q$) denotes the sheaf of real (resp.\ real
$d$-closed) $q$-forms on $M$, the short exact sequences
$$
\cmalign{ 0&\longrightarrow\cZ^1&\longrightarrow
\cE^1&\buildo d\over\longrightarrow &\cZ^2&\longrightarrow 0\cr
0&\longrightarrow\bR&\longrightarrow\cE^0&\buildo
d\over\longrightarrow &\cZ^1&\longrightarrow 0\cr}
$$

yield isomorphisms (with the sign convention of (IV-6.11))
$$\leqalignno{
&H^2_{DR}(M,\bR):=H^0(M,\cZ^2)/dH^0(M,\cE^1)\buildo -\partial
\over\longrightarrow H^1(M,\cZ^1),\phantom{\Big)}&(9.6)\cr
&H^1(M,\cZ^1)\buildo\partial\over\longrightarrow H^2(M,\bR).&(9.7)\cr}$$
Formula 3.4 gives $A_\beta=A_\alpha+g^{-1}_{\alpha\beta}dg_{\alpha\beta}$.
Since $\Theta(D)=dA_{\alpha}$ on $V_\alpha$, the image of $\{{\ii\over 2\pi}\Theta(D)\}$
under (9.6) is the \v Cech 1-cocycle with values in $\cZ^1$
$$\Big\{-{\ii\over 2\pi}(A_\beta-A_\alpha)\Big\}=\Big\{{1\over 2\pi\ii}
g^{-1}_{\alpha\beta}dg_{\alpha\beta}\Big\}=\{du_{\alpha\beta}\}$$
and the image of this cocycle under (9.7) is the 
\v Cech 2-cocycle $\{\delta u\}$ in $H^2(M,\bR)$. But $\{\delta u\}$
is precisely the image of $c_1(E)\in H^2(M,\bZ)$ in $H^2(M,\bR)$.\qed
\endproof

Let us assume now that $M$ is oriented and that $s\in \cC^\infty(M,E)$ is
transverse to the zero section of $E$, i.e.\ that $Ds\in\Hom(TM,E)$ is
surjective at every point of the zero set $Z:=s^{-1}(0)$. Then $Z$ is an
oriented 2-codi\-mensional submanifold of $M$ (the orientation of $Z$ is
uniquely defined by those of $M$ and $E$). We denote by $[Z]$ the current of 
integration over $Z$ and by $\{[Z]\}\in H^2_{DR}(M,\bR)$ its cohomology class.

\claim{(9.8) Theorem} We have $\{[Z]\}=c_1(E)_\bR$.
\endclaim

\proof{Proof} Consider the differential 1-form
$$u=s^{-1}\otimes Ds\in \cC^\infty_1(M\ssm Z,\bC).$$
Relatively to any trivialization $\theta$ of $E_{\restriction\Omega}$, one has $D_{\restriction
\Omega}\simeq_\theta d+A\wedge\bu$, thus
$$u_{\restriction\Omega}={d\sigma\over\sigma}+A~~~\hbox{\rm where}~~
\sigma=\theta(s).$$
It follows that $u$ has locally integrable coefficients on $M$. If 
$d\sigma/\sigma$ is considered as a current on $\Omega$, then
$$d\Big({d\sigma\over\sigma}\Big)=d\Big(\sigma^\star{dz\over z}\Big)=
\sigma^\star d\Big({dz\over z}\Big)=\sigma^\star(2\pi\ii\delta_0)=2\pi\ii[Z]$$
because of the Cauchy residue formula (cf.\ Lemma I-2.10) and because $\sigma$
is a submersion in a neighborhood of $Z$ (cf.\ (I-1.19)). Now, we have 
$dA=\Theta(D)$ and Th.~9.8 follows from the resulting equality:
\medskip\noindent{(9.9)}\hfill$du=2\pi\ii\,[Z]+\Theta(E)$.\hfill$\square$
\endproof

\section{10}{Connections of Type (1,0) and (0,1) over Complex Manifolds}
Let $X$ be a complex manifold, $\dim_\bC X = n$ and $E$ a $\cC^\infty$ vector bundle 
of rank $r$ over $X$~; here, $E$ is not assumed to be holomorphic. We denote
by $\cC^\infty_{p,q}(X,E)$ the space of $\cC^\infty$ sections of the bundle
$\Lambda^{p,q}T^\star X\otimes E$. We have therefore a direct sum decomposition
$$\cC^\infty_l(X,E)=\bigoplus_{p+q=l}\cC^\infty_{p,q}(X,E).$$
Connections of type $(1,0)$ or $(0,1)$ are operators acting on vector valued
forms, which imitate the usual operators $d',d''$ acting on $\cC^\infty_{p,q}(X,\bC)$.
More precisely, a connection of type (1,0) on $E$ is a differential
operator $D'$ of order 1 acting on $\cC^\infty_{\bu,\bu}(X,E)$ and satisfying
the following two properties:
$$\leqalignno{
&D': \cC^\infty_{p,q}(X,E) \longrightarrow \cC^\infty_{p+1,q}(X,E),&(10.1)\cr
&D'(f\wedge s) = d'f \wedge s + (-1)^{\deg f}f\wedge D's&(10.1')\cr}$$
for any $f\in \cC^\infty_{p_1,q_1}(X,\bC),~ s\in \cC^\infty_{p_2,q_2}(X,E)$.
The definition of a connection $D''$ of type (0,1) is similar. If
$\theta: E_{\restriction\Omega} \to \Omega \times \bC^r$ is a $\cC^\infty$ trivialization of $E_{\restriction\Omega}$ 
and if $\sigma=(\sigma_\lambda) = \theta(s)$, then all such connections
$D'$ and $D''$ can be written
$$\leqalignno{
D's&\simeq_\theta d'\sigma + A'\wedge \sigma,&(10.2')\cr
D''s&\simeq_\theta d''\sigma + A''\wedge \sigma~&(10.2'')\cr}$$
where $A'\in \cC^\infty_{1,0}\big(\Omega,\Hom(\bC^r,\bC^r)\big),~
A'' \in \cC^\infty_{0,1}\big(\Omega,\Hom (\bC^r,\bC^r)\big)$
are arbitrary forms with matrix coefficients.

It is clear that $D = D'+D''$ is then a connection in the sense of \S 2~;
conversely any connection $D$ admits a unique decomposition $D = D'+D''$
in terms of a (1,0)-connection and a (0,1)-connection.

Assume now that $E$ has a hermitian structure and that $\theta$ is an
{\it isometry}. The connection $D$ is hermitian if and only if the connection 
form $A=A'+ A''$ satisfies $A^\star=-A$, and this condition is equivalent 
to $A'=-(A'')^\star$. From this observation, we get immediately:

\claim{(10.3) Proposition} Let $D''_0$ be a given $(0,1)$-connection on a
hermitian bundle\break \hbox{$\pi:E\to X$}. Then there exists a unique
hermitian connection
\hbox{$D = D'+D''$} such that $D''=D''_0$.
\endclaim

\section{11}{Holomorphic Vector Bundles}
From now on, the vector bundles $E$ in which we are interested are
supposed to have a {\it holomorphic structure\/}:

\claim{(11.1) Definition} A vector bundle $\pi: E \to X$ is said to be holomorphic
if $E$ is a complex manifold, if the projection map $\pi$ is holomorphic and
if there exists a covering $(V_\alpha)_{\alpha\in I}$ of $X$ and a family of
holomorphic trivializations $\theta_\alpha: E_{\restriction V_\alpha} \to
V_\alpha \times \bC^r$.
\endclaim

It follows that the transition matrices $g_{\alpha\beta}$ are holomorphic
on $V_\alpha \cap V_\beta$. In complete analogy with the discussion
of \S 8, we see that the concept of holomorphic vector bundle is equivalent
to the concept of locally free sheaf of modules over the ring $\cO$ of
germs of holomorphic functions on $X$. We shall denote by $\cO(E)$ the
associated sheaf of germs of holomorphic sections of $E$. In the case 
$r=1$, there is a one-to-one correspondence between the isomorphism classes 
of holomorphic line bundles and the \v Cech cohomology group 
$H^1(X,\cO^\star)$.

\claim{(11.2) Definition} The group $H^1(X,\cO^\star)$ of isomorphism 
classes of holomorphic line bundles is called the {\it Picard group} of $X$.
\endclaim

If $s\in \cC^\infty_{p,q}(X,E)$, the components
$\sigma^\alpha = (\sigma^\alpha_\lambda)_{1\le \lambda\le r} = \theta_\alpha(s)$
of $s$ under $\theta_\alpha$ are related by
$$\sigma^\alpha  = g_{\alpha\beta}\cdot\sigma^\beta \quad\hbox{\rm on}\quad
V_\alpha\cap V_\beta.$$
Since $d''g_{\alpha\beta} = 0$, it follows that
$$d''\sigma^\alpha = g_{\alpha\beta}\cdot d''\sigma^\beta\quad\hbox{\rm on}
\quad V_\alpha\cap V_\beta.$$
The collection of forms $(d''\sigma^\alpha)$ therefore corresponds to a
unique global 
$(p,q+1)$-form $d''s$ such that $\theta_\alpha(d''s)=d''\sigma^\alpha$, 
and the operator $d''$ defined in this way is a $(0,1)$-connection on $E$.

\claim{(11.3) Definition} The operator $d''$ is called the canonical 
$(0,1)$-connection of the holomorphic bundle $E$.
\endclaim

It is clear that  $d^{\prime\prime 2} = 0$. Therefore, for any integer 
$p=0,1,\ldots,n$, we get a complex
$$\cC^\infty_{p,0}(X,E) \buildo d''\over \longrightarrow \cdots \longrightarrow \cC^\infty_{p,q}(X,E)
\buildo d''\over \longrightarrow \cC^\infty_{p,q+1}(X,E)\longrightarrow\cdots$$
known as the {\it Dolbeault complex} of $(p,\bu)$-forms with values in $E$.

\claim{(11.4) Notation} The $q$-th cohomology group of the Dolbeault
complex is denoted $H^{p,q}(X,E)$ and is called the $(p,q)$ Dolbeault
cohomology group with values in~$E$.
\endclaim

The Dolbeault-Grothendieck lemma I-2.11 shows that the complex of sheaves
$d'':\cC^\infty_{0,\bu}(X,E)$ is a soft resolution of the sheaf
$\cO(E)$. By the De Rham-Weil isomorphism theorem IV-6.4, we get:

\claim{(11.5) Proposition} $H^{0,q}(X,E)\simeq H^q\big(X,\cO(E)\big)$.
\endclaim

Most often, we will identify the locally free sheaf $\cO(E)$ and the
bundle $E$ itself~; the above sheaf cohomology group will therefore be simply
denoted $H^q(X,E)$. Another standard notation in analytic or algebraic
geometry~is:

\claim{(11.6) Notation} If $X$ is a complex manifold, $\Omega^p_X$
denotes the vector bundle $\Lambda^pT^\star X$ or its sheaf of sections.
\endclaim

It is clear that the complex $\cC^\infty_{p,\bu}(X,E)$ is identical to
the complex $\cC^\infty_{0,\bu}(X,\Omega^p_X\otimes E)$, therefore
we obtain a canonical isomorphism:

\claim{(11.7) Dolbeault isomorphism} 
$H^{p,q}(X,E)\simeq H^q(X,\Omega^p_X\otimes E)$.
\endclaim
In particular, $H^{p,0}(X,E)$ is the space of global holomorphic 
sections of the bundle $\Omega^p_X\otimes E$.

\section{12}{Chern Connection}
Let $\pi: E\to X$ be a {\it hermitian holomorphic} vector bundle of rank $r$.
By Prop.~10.3, there exists a unique hermitian connection $D$ such that
\hbox{$D''= d''$.}

\claim{(12.1) Definition} The unique hermitian connection~ $D$~ such that
$D''=d''$ is called the Chern connection of $E$. The curvature tensor of this
connection will be denoted by $\Theta(E)$ and is called the Chern curvature
tensor of $E$.
\endclaim

Let us compute  $D$ with respect to an arbitrary {\it holomorphic 
trivialization} $\theta: E_{\restriction\Omega} \to \Omega\times \bC^r$. 
Let $H=(h_{\lambda\mu})_{1\le \lambda,\mu\le r}$ denote the hermitian matrix 
with $\cC^\infty$ coefficients representing the metric along the fibers of 
$E_{\restriction\Omega}$. For any $s,t \in \cC^\infty_{\bu,\bu}(X,E)$ and
$\sigma=\theta(s),~\tau = \theta(t)$ one can write
$$\{s,t\}=\sum_{\lambda,\mu}h_{\lambda\mu}\sigma_\lambda \wedge
\ol\tau_\mu=\sigma^\dagger\wedge H\ol\tau,$$
where $\sigma^\dagger$ is the transposed matrix of $\sigma$. It follows that
$$\eqalign{
\{ Ds,t\} + &(-1)^{\deg s} \{ s,Dt\} = d\{ s,t\}\cr
&= (d\sigma)^\dagger \wedge H\ol \tau + (-1)^{\deg \sigma} \sigma^\dagger \wedge (dH\wedge \ol \tau + H\ol{d\tau})\cr
&= \big(d\sigma +\ol H^{-1}d'\ol H \wedge \sigma\big)^\dagger \wedge H\ol
 \tau + (-1)^{\deg \sigma} \sigma^\dagger \wedge (\ol{d\tau+ \ol H^{-1}d'\ol H \wedge \tau)}\cr}$$
using the fact that $dH = d'H + \ol{d'\ol H}$ and
$\ol H ^\dagger = H$. Therefore the Chern connection $D$ coincides with the
hermitian connection defined by
$$\leqalignno{
Ds&{}\simeq_\theta d\sigma+\ol H^{-1}d'\ol H \wedge \sigma,&(12.2)\cr
D'&{}\simeq_\theta d'+\ol H^{-1} d'\ol H\wedge\bu=
\ol H^{-1} d'(\ol H\bu),~~~~D'' = d''.&(12.3)\cr}$$
It is clear from this relations that $D^{\prime 2}=D^{\prime\prime 2}=0$.
Consequently $D^2$ is given by to $D^2=D'D''+D''D'$, and the curvature 
tensor $\Theta(E)$ is of type~$(1,1)$. Since $d'd''+d''d'=0$, we get
$$(D'D''+D''D')s\simeq_\theta\ol H^{-1}d'\ol H\wedge d''\sigma 
+d''(\ol H^{-1}d'\ol H\wedge\sigma)=d''(\ol H^{-1}d'\ol H)\wedge\sigma.$$

\claim{(12.4) Theorem} The Chern curvature tensor is such that
$$\ii\,\Theta(E) \in \cC^\infty_{1,1}(X,\Herm(E,E)).$$ If $\theta:
E_{\restriction\Omega}\to\Omega\times\bC^r$ is a holomorphic
trivialization and  if $H$ is the hermitian matrix representing the
metric along the fibers of  $E_{\restriction\Omega}$, then
$$\ii\,\Theta(E)=\ii\,d''(\ol H^{-1} d'\ol H)\quad\hbox{\rm on }\quad
\Omega.$$
\endclaim

Let $(e_1,\ldots,e_r)$ be a $\cC^\infty$ orthonormal frame of $E$ over a coordinate
patch $\Omega \subset X$ with complex coordinates $(z_1,\ldots,z_n)$. On $\Omega$
the Chern curvature tensor can be written
$$\ii \Theta(E)=\ii\sum_{1\le j,k\le n,~1\le \lambda,\mu\le r}
c_{jk\lambda\mu}\,dz_j \wedge d\ol z_k \otimes e^\star_\lambda \otimes e_\mu
\leqno(12.5)$$
for some coefficients $c_{jk\lambda\mu} \in \bC$. The hermitian property
of $\ii \Theta(E)$ means that $\ol c_{jk\lambda\mu} = c_{kj\mu\lambda}$.

\claim{(12.6) Special case} \rm When
$r=\hbox{\rm rank~}E=1$, the hermitian matrix $H$
is a positive function which we write $H=e^{-\varphi}$, $\varphi\in 
\cC^\infty(\Omega,\bR)$. By the above formulas we get
$$D'\simeq_\theta d'-d'\varphi\wedge{\bu}=e^\varphi d'(e^{-\varphi}\bu),
\leqno(12.7)$$
$$\ii \Theta(E)=\ii d'd''\varphi\quad\hbox{\rm on }\quad \Omega.\leqno (12.8)$$
Especially, we see that $\ii\,\Theta(E)$ is a {\it closed} real (1,1)-form
on $X$.
\endclaim

\claim{(12.9) Remark} \rm In general, it is not possible to find local
frames $(e_1,\ldots,e_r)$ of $E_{\restriction\Omega}$ that are simultaneously
{\it holomorphic} and {\it orthonormal}. In fact, we have in this case
$H=(\delta_{\lambda\mu})$, so a necessary condition for the existence of
such a frame is that $\Theta(E)=0$ on $\Omega$. Conversely, if $\Theta(E)=0$, 
Prop.~6.7 and Rem.~7.5 show that $E$ possesses local
orthonormal parallel frames $(e_\lambda)$~; we have in particular
$D''e_\lambda=0$, so $(e_\lambda)$ is holomorphic; such a bundle
$E$ arising from a unitary representation of $\pi_1(X)$ is said
to be {\it hermitian flat}. The next proposition shows in a
more local way that the Chern curvature tensor is the
obstruction to the existence of orthonormal holomorphic frames:
a holomorphic frame can be made ``almost orthonormal" only up to
curvature terms of order $2$ in a neighborhood of any point.
\endclaim

\claim{(12.10) Proposition} For every point $x_0\in X$ and every
coordinate system $(z_j)_{1\le j\le n}$ at $x_0$, there exists a
holomorphic frame $(e_\lambda)_{1\le\lambda\le r}$ in a neighborhood of
$x_0$ such that
$$\langle e_\lambda(z),e_\mu(z)\rangle=\delta_{\lambda\mu}- 
\sum_{1\le j,k\le n}c_{jk\lambda\mu}\,z_j\ol z_k+O(|z|^3)$$ 
where $(c_{jk\lambda\mu})$ are the coefficients of the Chern curvature
tensor $\Theta(E)_{x_0}$. Such a frame $(e_\lambda)$ is called a normal
coordinate frame at~$x_0$.
\endclaim

\proof{Proof} Let $(h_\lambda)$ be a holomorphic frame of $E$. After replacing
$(h_\lambda)$ by suitable linear combinations with constant
coefficients, we may assume that $\big(h_\lambda(x_0)\big)$ is an
orthonormal basis of $E_{x_0}$.  Then the inner products 
$\langle h_\lambda,h_\mu\rangle$ have an expansion 
$$\langle h_\lambda(z),h_\mu(z)\rangle=\delta_{\lambda\mu}+
\sum_j(a_{j\lambda\mu}\,z_j+a'_{j\lambda\mu}\,\ol z_j)+O(|z|^2)$$
for some complex coefficients $a_{j\lambda\mu}$, $a'_{j\lambda\mu}$ such
that $a'_{j\lambda\mu}=\ol a_{j\mu\lambda}$.  Set first
$$g_\lambda(z)=h_\lambda(z)-\sum_{j,\mu}a_{j\lambda\mu}\,z_j\,h_\mu(z).$$
Then there are coefficients $a_{jk\lambda\mu}$, $a'_{jk\lambda\mu}$,
$a''_{jk\lambda\mu}$ such that
$$\eqalign{
\langle g_\lambda(z),g_\mu(z)\rangle&=\delta_{\lambda\mu}+O(|z|^2)\cr
&=\delta_{\lambda\mu}+\sum_{j,k}\big(a_{jk\lambda\mu}\,z_j\ol z_k+
a'_{jk\lambda\mu}\,z_jz_k+a''_{jk\lambda\mu}\ol z_j\ol z_k\big)+
O(|z|^3).}$$ 
The holomorphic frame $(e_\lambda)$ we are looking for is 
$$e_\lambda(z)=
g_\lambda(z)-\sum_{j,k,\mu}a'_{jk\lambda\mu}\,z_jz_k\,g_\mu(z).$$
Since $a''_{jk\lambda\mu}=\ol a'_{jk\mu\lambda}$, we easily find
$$\eqalign{
\langle e_\lambda(z),e_\mu(z)\rangle&=\delta_{\lambda\mu}+
\sum_{j,k}a_{jk\lambda\mu}\,z_j\ol z_k+O(|z|^3),\cr
d'\langle e_\lambda,e_\mu\rangle&=\{D'e_\lambda,e_\mu\}=
\sum_{j,k}a_{jk\lambda\mu}\,\ol z_k\,dz_j+O(|z|^2),\cr
\Theta(E)\cdot e_\lambda
&=D''(D'e_\lambda)=\sum_{j,k,\mu}a_{jk\lambda\mu}\,d\ol z_k\wedge dz_j
\otimes e_\mu+O(|z|),\cr}$$ 
therefore $c_{jk\lambda\mu}=-a_{jk\lambda\mu}$.\qed
\endproof

\section{13}{Lelong-Poincar\'e Equation and First Chern Class}
Our goal here is to extend the Lelong-Poincar\'e equation III-2.15
to any meromorphic section of a holomorphic line bundle.

\claim{(13.1) Definition} A meromorphic section of a bundle $E\to X$ is
a section $s$ defined on an open dense subset of $X$, such that for
every trivialization $\theta_\alpha:E_{\restriction V_\alpha}\to
V_\alpha\times\bC^r$ the components of
$\sigma^\alpha=\theta_\alpha(s)$ are meromorphic functions on $V_\alpha$.
\endclaim
 
Let $E$ be a hermitian line bundle, $s$ a meromorphic section which does not
vanish on any component of $X$ and $\sigma=\theta(s)$ the corresponding
meromorphic function in a trivialization $\theta:E_{\restriction \Omega}\to
\Omega\times\bC$. The divisor of $s$ is the current on $X$ defined by
$\div\,s_{\restriction\Omega}=\div\,\sigma$ for all trivializing
open sets $\Omega$. One can write $\div\,s=\sum m_jZ_j$, where
the sets $Z_j$ are the irreducible components of the sets of zeroes and poles 
of $s$ (cf.\ \S~II-5). The Lelong-Poincar\'e equation (II-5.32) gives
$${\ii\over \pi} d'd'' \log|\sigma| = \sum m_j[Z_j],$$
and from the equalities $|s|^2 = |\sigma|^2 e^{-\varphi}$ and $d'd''\varphi=
\Theta(E)$ we get
$$id'd''\log|s|^2 = 2\pi\sum m_j[Z_j] -\ii\,\Theta(E). \leqno(13.2)$$
This equality can be viewed as a complex analogue of (9.9) (except that here
the hypersurfaces $Z_j$ are not necessarily smooth).
In particular, if $s$ is a {\it non vanishing holomorphic} section of
$E_{\restriction\Omega}$, we have
$$\ii\,\Theta(E)=-\ii d'd''\log|s|^2~~~~\hbox{\rm on}~~\Omega.\leqno (13.3)$$

\claim{(13.4) Theorem} Let $E\to X$ be a line bundle and let $s$ be a meromorphic 
section of $E$ which does not vanish identically on any component of $X$.
If~$\sum m_jZ_j$ is the divisor of $s$, then
$$c_1(E)_\bR=\Big\{\sum m_j[Z_j]\Big\}\in H^2(X,\bR).$$
\endclaim

\proof{Proof} Apply Formula (13.2) and Th.~9.5, and observe that the
bidimension $(1,1)$-current
$\ii d'd''\log|s|^2=d\big(\ii d''\log|s|^2\big)$ has zero cohomology class.\qed
\endproof

\claim{(13.5) Example} \rm If $\Delta=\sum m_jZ_j$ is an arbitrary divisor
on $X$, we associate to $\Delta$ the sheaf $\cO(\Delta)$ of germs of 
meromorphic functions $f$ such that \hbox{$\div(f)+\Delta\ge 0$.} Let 
$(V_\alpha)$ be a covering of $X$ and $u_\alpha$ a meromorphic function on
$V_\alpha$ such that $\div(u_\alpha)=\Delta$ on $V_\alpha$. Then 
$\cO(\Delta)_{\restriction V_\alpha}=u_\alpha^{-1}\cO$, thus
$\cO(\Delta)$ is a locally free $\cO$-module of rank~1. This sheaf
can be identified to the line bundle $E$ over $X$ defined by the cocycle
$g_{\alpha\beta}:=u_\alpha/u_\beta\in\cO^\star(V_\alpha\cap V_\beta)$. 
In fact, there is a sheaf isomorphism $\cO(\Delta)\longrightarrow\cO(E)$ defined by
$$\cO(\Delta)(\Omega)\ni f\longmapsto s\in \cO(E)(\Omega)~~
\hbox{\rm with}~~\theta_\alpha(s)=fu_\alpha~~
\hbox{\rm on}~\Omega\cap V_\alpha.$$
The constant meromorphic function $f=1$ induces a meromorphic
section $s$ of $E$ such that $\div\,s=\div\,u_\alpha=\Delta$~;
in the special case when $\Delta\ge 0$, the section $s$ is holomorphic
and its zero set $s^{-1}(0)$ is the support of $\Delta$.
By Th.~13.4, we have 
$$c_1\big(\cO(\Delta)\big)_\bR=\{[\Delta]\}.\leqno(13.6)$$
Let us consider the exact sequence $1\to\cO^\star\to\cM^\star\to\Div\to0$
already described in (II-5.36). There is a corresponding cohomology exact
sequence
$$\cM^\star(X)\longrightarrow\Div(X)\buildo\partial^0\over\longrightarrow H^1(X,\cO^\star).
\leqno(13.7)$$
The connecting homomorphism $\partial^0$ is equal to the map
$$\Delta\longmapsto\hbox{\rm isomorphism class of}~~\cO(\Delta)$$
defined above. The kernel of this map consists of divisors which are
divisors of global meromorphic functions in $\cM^\star(X)$. In
particular, two divisors $\Delta_1$ and $\Delta_2$ give rise to
isomorphic line bundles $\cO(\Delta_1)\simeq\cO(\Delta_2)$ if and
only if $\Delta_2-\Delta_1=\div(f)$ for some global
meromorphic function $f\in\cM^\star(X)$~; such divisors are called
{\it linearly equivalent}. The image
of $\partial^0$ consists of classes of line bundles $E$ such that
$E$ has a global meromorphic section which does not vanish on any component
of $X$. Indeed, if $s$ is such a section and $\Delta=\div\,s$,
there is an isomorphism 
\medskip\noindent{(13.8)}\hfill$\cO(\Delta)\buildo\simeq\over\longrightarrow\cO(E),~~~~
f\longmapsto fs$.\hfill$\square$
\endclaim

The last result of this section is a characterization of 
2-forms on $X$ which can be written as the curvature form of a hermitian
holomorphic line bundle.

\claim{(13.9) Theorem} Let $X$ be an arbitrary complex manifold.
\medskip
\item{\rm a)} For any hermitian line bundle $E$ over $M$, the Chern
curvature form ${\ii\over 2\pi}\Theta(E)$ is a closed real $(1,1)$-form
whose De Rham cohomology class is the image of an integral class.
\medskip
\item{\rm b)} Conversely, let $\omega$ be a $\cC^\infty$ closed real 
$(1,1)$-form such that
$\{\omega\}\in H^2_{DR}(X,\bR)$ is the image of an integral class. Then
there exists a hermitian line bundle $E\to X$ such that ${\ii\over 2\pi}\Theta(E)=
\omega$.\smallskip
\endclaim

\proof{Proof} a) is an immediate consequence of Formula (12.9) and Th.~9.5, 
so we have only to prove the converse part b). By Prop.~III-1.20, there 
exist an open covering
$(V_\alpha)$ of $X$ and functions $\varphi_\alpha\in \cC^\infty(V_\alpha,\bR)$
such that ${\ii\over 2\pi}d'd''\varphi_\alpha=\omega$ on $V_\alpha$.
It follows that the function $\varphi_\beta-\varphi_\alpha$ 
is pluriharmonic on $V_\alpha\cap V_\beta$. If $(V_\alpha)$ is chosen
such that the intersections $V_\alpha\cap V_\beta$ are simply connected, 
then Th.~I-3.35 yields holomorphic functions $f_{\alpha\beta}$ on 
$V_\alpha\cap V_\beta$ such that
$$2\Re f_{\alpha\beta}=\varphi_\beta-\varphi_\alpha~~~~\hbox{\rm on}~~
V_\alpha\cap V_\beta.$$
Now, our aim is to prove (roughly speaking) that $\big(\exp(-f_{\alpha\beta})
\big)$ is a cocycle in $\cO^\star$ that defines the line bundle $E$ we are 
looking for. The \v Cech differential $(\delta f)_{\alpha\beta\gamma}=
f_{\beta\gamma}-f_{\alpha\gamma}+f_{\alpha\beta}$ takes values in 
the constant sheaf $\ii\bR$ because
$$2\Re\,(\delta f)_{\alpha\beta\gamma}=(\varphi_\gamma-\varphi_\beta)-
(\varphi_\gamma-\varphi_\alpha)+(\varphi_\beta-\varphi_\alpha)=0.$$
Consider the real 1-forms $A_\alpha={\ii\over 4\pi}(d''\varphi_\alpha-
d'\varphi_\alpha)$. As $d'(\varphi_\beta-\varphi_\alpha)$ is equal to
$d'(f_{\alpha\beta}+\ol f_{\alpha\beta})=df_{\alpha\beta}$, we get
$$(\delta A)_{\alpha\beta}=A_\beta-A_\alpha=
{\ii\over 4\pi}d(\ol f_{\alpha\beta}-f_{\alpha\beta})=
{1\over 2\pi}d\Im f_{\alpha\beta}.$$
Since $\omega=dA_\alpha$, it follows by (9.6) and (9.7) that the 
\v Cech cohomology class {$\{\delta({1\over2\pi}\!\Im f_{\alpha\beta})\}
\kern-1pt$}
is equal to $\{\omega\}\in H^2(X,\bR)$, which is by 
hypothesis the image of a 2-cocycle $(n_{\alpha\beta\gamma})\in H^2(X,\bZ)$.
Thus we can write
$$\delta\Big({1\over2\pi}\Im f_{\alpha\beta}\Big)=
(n_{\alpha\beta\gamma})+\delta(c_{\alpha\beta})$$
for some 1-chain $(c_{\alpha\beta})$ with values in $\bR$. If we replace
$f_{\alpha\beta}$ by $f_{\alpha\beta}-2\pi\ii c_{\alpha\beta}$, then we
can achieve $c_{\alpha\beta}=0$, so $\delta(f_{\alpha\beta})\in
2\pi\ii\bZ$ and $g_{\alpha\beta}:=\exp(-f_{\alpha\beta})$ will be a cocycle 
with values in $\cO^\star$. Since
 $$\varphi_\beta-\varphi_\alpha=2\Re f_{\alpha\beta}=
   -\log|g_{\alpha\beta}|^2,$$
the line bundle $E$ associated to this cocycle admits a global hermitian
metric defined in every trivialization by the matrix
$H_\alpha=(\exp(-\varphi_\alpha))$ and therefore 
$${\ii\over 2\pi}\Theta(E)={\ii\over 2\pi}d'd''\varphi_\alpha=\omega~~~~
\hbox{\rm on}~~V_\alpha.\eqno{\square}$$
\endproof

\section{14}{Exact Sequences of Hermitian Vector Bundles}
Let us consider an exact sequence of holomorphic vector bundles over $X\,$:
$$0 \longrightarrow S \buildo j\over \longrightarrow E \buildo g\over \longrightarrow Q \longrightarrow 0.\leqno(14.1)$$
Then $E$ is said to be an {\it extension of $S$ by $Q$}.
A (holomorphic, resp.\ $\cC^\infty$) splitting of the exact sequence is a
(holomorphic, resp.\ $\cC^\infty$) homomorphism $h:Q\longrightarrow E$ which is a right inverse
of the projection $E\longrightarrow Q$, i.e.\ such that $g\circ h=\Id_Q$.

Assume that a $\cC^\infty$ hermitian metric on $E$ is given. Then $S$ and $Q$
can be endowed with the induced and quotient metrics respectively. Let us
denote by $D_E,~D_S,~D_Q$ the corresponding Chern connections. The
adjoint homomorphisms
$$j^\star: E\longrightarrow S,~~~~g^\star: Q\longrightarrow E$$
are $\cC^\infty$ and can be described respectively as the orthogonal projection 
of $E$ onto $S$ and as the orthogonal splitting of the exact sequence (14.1).
They yield a $\cC^\infty$ (in general non analytic) isomorphism 
$$j^\star \oplus g: E \buildo \simeq \over \longrightarrow S\oplus Q.\leqno(14.2)$$

\claim{(14.3) Theorem} According to the $\cC^\infty$ isomorphism $j^\star\oplus g$,
$D_E$ can be written
$$D_E = \pmatrix{D_S&-\beta^\star\cr\beta & D_Q\cr}$$
where $\beta \in \cC^\infty_{1,0}\big(X,\Hom(S,Q)\big)$ is called the
second fundamental of $S$ in $E$ and where $\beta^\star \in 
\cC^\infty_{0,1}\big(X,\Hom(Q,S)\big)$ is the adjoint of $\beta$. 
Furthermore, the following identities hold:
\medskip\noindent
$\cmalign{{\rm a)}\quad&D'_{\Hom(S,E)}j&=g^\star\circ\beta,
                   &~~d''j&=0~;\cr
          {\rm b)}\quad&D'_{\Hom(E,Q)}g&=-\beta\circ j^\star,
                   &~~d''g&=0~;\cr
          {\rm c)}\quad&D'_{\Hom(E,S)}j^\star&=0,
                   &~~d''j^\star&=\beta^\star\circ g~;\cr
          {\rm d)}\quad&D'_{\Hom(Q,E)}g^\star&=0,
                   &~~d''g^\star&=-j\circ\beta^\star.\cr}$
\endclaim                   

\proof{Proof} If we define $\nabla_E \simeq D_S \oplus D_Q$ via (14.2),
then  $\nabla_E$ is a hermitian connection on $E$. By (7.3), we have therefore
$D_E = \nabla_E +\Gamma\wedge\bu$, where $\Gamma\in \cC^\infty_1(X,\Hom(E,E))$ 
and $\Gamma^\star = -\Gamma$. Let us write
$$\Gamma={\alpha~~\gamma\choose\beta\,~~\delta},~~~~
\alpha^\star = -\alpha,~\delta^\star = -\delta,~\gamma = -\beta^\star,$$
$$D_E = \pmatrix{
D_S+\alpha &\gamma\cr
\beta& D_Q+\delta\cr}.\leqno(14.4)$$
For any section $u\in \cC^\infty_{\bu,\bu}(X,E)$ we have
$$\eqalign{
D_Eu &= D_E(jj^\star u{+}g^\star gu)\cr
&= j D_S(j^\star u){+}g^\star D_Q(gu){+}(D_{\Hom(S,E)}j){\wedge}j^\star u{+}
(D_{\Hom(E,Q)}g^\star){\wedge}gu.\cr}$$
A comparison with (14.4) yields
$$\eqalign{
D_{\Hom(S,E)}j &= j\circ \alpha + g^\star \circ \beta,\cr
D_{\Hom(E,Q)}g^\star &= j\circ \gamma + g^\star \circ \delta,\cr}$$
Since $j$ is holomorphic, we have $d''j = j\circ \alpha^{0,1} + g^\star
\circ \beta^{0,1}=0$, thus $\alpha^{0,1} = \beta^{0,1} = 0$.  But
$\alpha^\star = -\alpha$, hence $\alpha=0$ and $\beta\in
\cC^\infty_{1,0}(\Hom (S,Q))$ ; identity a) follows.  Similarly, we get
$$\eqalign{ D_S(j^\star u) &= j^\star D_Eu +
(D_{\Hom(E,S)}j^\star)\wedge u,\cr
D_Q(g u) &= g D_Eu + (D_{\Hom(E,Q)}g)\wedge u,\cr}$$ 
and comparison with (14.4) yields
$$\eqalign{
D_{\Hom(E,S)}j^\star &= -\alpha\circ j^\star -\gamma\circ g
= \beta^\star\circ g,\cr
D_{\Hom(E,Q)}g &= -\beta\circ j^\star -\delta\circ g.\cr}$$ 
Since $d''g = 0$, we get $\delta^{0,1} = 0$, hence $\delta=0$. 
Identities b), c), d) follow from the above computations.\qed
\endproof

\claim{(14.5) Theorem} We have $d''(\beta^\star)=0$, and the Chern curvature
of $E$ is 
$$\Theta(E)=\pmatrix{
\Theta(S)-\beta^\star \wedge \beta & D'_{\Hom(Q,S)}\beta^\star\cr
d''\beta &\Theta(Q)-\beta\wedge \beta^\star\cr}.$$
\endclaim

\proof{Proof} A computation of $D^2_E$ yields
$$D^2_E = \pmatrix{
D^2_S-\beta^\star \wedge \beta &-(D_S\circ\beta^\star+\beta^\star\circ D_Q)\cr
\beta\circ D_S + D_Q\circ \beta &D^2_Q-\beta\wedge \beta^\star\cr}.$$
Formula (13.4) implies
$$\eqalign{
D_{\Hom(S,Q)}\beta&= \beta\circ D_S +D_Q\circ \beta,\cr
D_{\Hom(Q,S)}\beta^\star&= D_S\circ \beta^\star +\beta^\star\circ D_Q .\cr}$$
Since $D^2_E$ is of type (1,1), it follows that $d''\beta^\star = D''_
{\Hom(Q,S)}\beta^\star=0$. The proof is achieved.\qed
\endproof

A consequence of Th.~14.5 is that $\Theta(S)$ and $\Theta(Q)$ are
given in terms of $\Theta(E)$ by the following formulas, where
$\Theta(E)_{\restriction S}$, $\Theta(E)_{\restriction Q}$ denote the
blocks in the matrix of $\Theta(E)$ corresponding to $\Hom(S,S)$ and
$\Hom(Q,Q)$:
$$\leqalignno{
\Theta(S)&= \Theta(E)_{\restriction S} + \beta^\star \wedge \beta,&(14.6)\cr
\Theta(Q)&= \Theta(E)_{\restriction Q} + \beta \wedge \beta^\star.&(14.7)\cr}$$
By 14.3 c) the second fundamental form $\beta$ vanishes identically 
if and only if the orthogonal splitting $E\simeq S\oplus Q$ is holomorphic~;
then we have $\Theta(E)=\Theta(S)\oplus \Theta(Q)$.
\medskip
\medskip
Next, we show that the $d''$-cohomology class $\{\beta^\star\}{\in}
H^{0,1}\big(X,\Hom(Q,S)\big)$ characterizes the isomorphism class
of $E$ among all extensions of $S$ by $Q$. Two extensions $E$
and $F$ are said to be isomorphic if there is a commutative diagram
of holomorphic maps
$$\cmalign{
0~\longrightarrow~ &S~\longrightarrow~ &E~\longrightarrow~ &Q~\longrightarrow~ 0\phantom{\Big)}\cr
&\big|\big|\hfill&\big\downarrow\hfill&\big|\big|\hfill\cr
0~\longrightarrow~ &S~\longrightarrow~ &F~\longrightarrow~ &Q~\longrightarrow~ 0\phantom{\Big)}\cr}\leqno(14.8)$$
in which the rows are exact sequences. The central vertical arrow is then 
necessarily an isomorphism.
It is easily seen that $0\to S\to E\to Q\to 0$ has a holomorphic splitting 
if and only if $E$ is isomorphic to the trivial extension $S\oplus Q$.

\claim{(14.9) Proposition} The correspondence
$$\{E\}\longmapsto\{\beta^\star\}$$
induces a bijection from the set of isomorphism classes of extensions of $S$ 
by $Q$ onto the cohomology group $H^1\big(X,\Hom(Q,S)\big)$. In
particular $\{\beta^\star\}$ vanishes if and only if the exact sequence 
$$0\longrightarrow S\buildo j\over\longrightarrow E\buildo g\over\longrightarrow Q\longrightarrow 0$$
splits holomorphically.
\endclaim

\proof{Proof} a) The map is well defined, i.e.\ $\{\beta^\star\}$ does not
depend on the choice of the hermitian metric on~$E$. Indeed, a new hermitian
metric produces a new $\cC^\infty$ splitting $\wh g^\star$ and a new
form $\wh\beta^\star$ such that $d''\wh g^\star=-j\circ\wh\beta^\star$.
Then $gg^\star=g\wh g^\star=\Id_Q$, thus $\wh g-g=j\circ v$ for
some section $v\in \cC^\infty\big(X,\Hom(Q,S)\big)$. It follows that
$\smash{\wh\beta}^\star-\beta^\star=-d''v$. Moreover, it is clear that
an isomorphic extension $F$ has the same associated form $\beta^\star$
if $F$ is endowed with the image of the hermitian metric of $E$.
\medskip
\noindent{b)} The map is injective. Let $E$ and $F$ be extensions of $S$ by $Q$.
Select $\cC^\infty$ splittings $E,F\simeq S\oplus Q$. We endow
$S,Q$ with arbitrary hermitian metrics and $E,F$ with the direct sum metric.
Then we have corresponding $(0,1)$-connections
$$D''_E = \pmatrix{D''_S&-\beta^\star\cr0 & D''_Q\cr},~~~~
D''_F = \pmatrix{D''_S&-\wt\beta^\star\cr0 & D''_Q\cr}.$$
Assume that $\wt\beta^\star=\beta^\star+d''v$ for some 
$v\in \cC^\infty\big(X,\Hom(Q,S)\big)$. The isomorphism $\Psi:E\longrightarrow F$ of class $\cC^\infty$
defined by the matrix
$$\pmatrix{\Id_S& v\cr0 & \Id_Q\cr}.$$
is then holomorphic, because the relation $D''_S\circ v-v\circ D''_Q=d''v=
\wt\beta^\star-\beta^\star$ implies
$$\eqalign{
D''_{\Hom(E,F)}\Psi&=D''_F\circ\Psi-\Psi\circ D''_E\cr
&=\pmatrix{D''_S&-\wt\beta^\star\cr0 & D''_Q\cr}
\pmatrix{\Id_S& v \cr0 & \Id_Q\cr}-
\pmatrix{\Id_S& v \cr0 & \Id_Q\cr}
\pmatrix{D''_S&-\beta^\star\cr0 & D''_Q\cr}\cr
&=\pmatrix{0&-\wt\beta^\star+\beta^\star+(D''_S\circ v-v\circ D''_Q)\cr
           0& 0\cr}=0.\cr}$$
Hence the extensions $E$ and $F$ are isomorphic.
\medskip
\noindent{c)} The map is surjective. Let $\gamma$ be an arbitrary $d''$-closed
$(0,1)$-form on $X$ with values in $\Hom(Q,S)$. We define $E$
as the $\cC^\infty$ hermitian vector bundle $S\oplus Q$ endowed with the 
$(0,1)$-connection
$$D''_E=\pmatrix{D''_S&\gamma\cr0 & D''_Q\cr}.$$
We only have to show that this connection is induced by a holomorphic
structure on $E$~; then we will have $\beta^\star=-\gamma$. However,
the Dolbeault-Grothendieck lemma implies that there is a covering of
$X$ by open sets $U_\alpha$ on which $\gamma=d''v_\alpha$ for
some $v_\alpha\in \cC^\infty\big(U_\alpha,\Hom(Q,S)\big)$. Part b) above
shows that the matrix
$$\pmatrix{\Id_S&v_\alpha\cr0 & \Id_Q\cr}$$
defines an isomorphism $\psi_\alpha$ from $E_{\restriction U_\alpha}$
onto the trivial extension $(S\oplus Q)_{\restriction U_\alpha}$
such that $D''_{\Hom(E,S\oplus Q)}\psi_\alpha=0$. The required
holomorphic structure on $E_{\restriction U_\alpha}$ is the inverse image
of the holomorphic structure of $(S\oplus Q)_{\restriction U_\alpha}$ by
$\psi_\alpha$~; it is independent of $\alpha$ because $v_\alpha-v_\beta$ and
$\psi_\alpha\circ\psi_\beta^{-1}$ are holomorphic on 
$U_\alpha\cap U_\beta$.\qed
\endproof

\claim{(14.10) Remark} \rm If $E$ and $F$ are extensions of $S$ by
$Q$ such that the corresponding forms $\beta^\star$ and 
$\wt\beta^\star=u\circ\beta^\star\circ v^{-1}$ differ by
$u\in H^0\big(X,\Aut(S)\big)$, $v\in H^0\big(X,\Aut(Q)\big)$,
it is easy to see that the bundles $E$ and $F$
are isomorphic. To see this, we need only replace the
vertical arrows representing the identity maps of $S$ and $Q$ in (14.8)
by $u$ and $v$ respectively. Thus, if we want
to classify isomorphism classes of bundles $E$ which are extensions of
$S$ by $Q$ rather than the extensions themselves, the set of classes is
the quotient of $H^1\big(X,\Hom(Q,S)\big)$ by the action of
$H^0\big(X,\Aut(S)\big)\times H^0\big(X,\Aut(Q)\big)$.
In particular, if $S,Q$ are line bundles and if $X$ is compact connected,
then $H^0\big(X,\Aut(S)\big)$, $H^0\big(X,\Aut(Q)\big)$
are equal to $\bC^\star$ and the set of classes is the projective space
$P\big(H^1(X,\Hom(Q,S))\big)$.
\endclaim

\section{15}{Line Bundles $\cO(k)$ over $\bP^n$}
\subsection{15.A}{Algebraic properties of $\cO(k)$}
Let $V$ be a complex vector space of dimension $n+1,~n\ge 1$. The
quotient topological space $P(V)=(V\ssm\{0\})/\bC^\star$ is called the
{\it projective space of $V$}, and can be considered as the set of lines in 
$V$ if $\{0\}$ is added to each class $\bC^\star\cdot x$. Let
$$\eqalign{\pi:~V\ssm\{0\}&\longrightarrow P(V)\cr
x&\longmapsto [x]=\bC^\star\cdot x\cr}$$
be the canonical projection. When $V=\bC^{n+1}$, we simply denote
$P(V)=\bP^n$. The space $\bP^n$ is the quotient $S^{2n+1}/S^1$ of
the unit sphere $S^{2n+1}\subset\bC^{n+1}$ by the multiplicative
action of the unit circle  $S^1\subset\bC$, so $\bP^n$ is compact.
Let $(e_0,\ldots,e_n)$ be a basis of $V$, and let $(x_0,\ldots,x_n)$ be the
coordinates of a vector $x\in V\ssm\{0\}$. Then $(x_0,\ldots,x_n)$ are
called the {\it homogeneous coordinates} of $[x]\in P(V)$. The space
$P(V)$ can be covered by the open sets $\Omega_j$ defined by
$\Omega_j=\{[x]\in P(V)\,;~x_j\ne 0\}$ and  there are homeomorphisms
$$\eqalign{
\tau_j~:~~\Omega_j&\longrightarrow\bC^n\cr
               [x]&\longmapsto(z_0,\ldots,\wh{z_j},\ldots,z_n),~~~~
z_l=x_l/x_j~~\hbox{\rm for}~~l\ne j.\cr}$$
The collection $(\tau_j)$ defines a holomorphic atlas on $P(V)$, thus
$P(V)=\bP^n$ is a compact $n$-dimensional complex analytic manifold. 

Let $\soul V$ be the trivial bundle $P(V)\times V$.
We denote by $\cO(-1)\subset\soul V$ the {\it tautological line subbundle}
$$\cO(-1)=\big\{([x],\xi)\in P(V)\times V~;~\xi\in\bC\cdot x\big\}\leqno(15.1)$$
such that $\cO(-1)_{[x]}=\bC\cdot x\subset V$, $x\in V\ssm\{0\}$.
Then $\smash{\cO(-1)_{\restriction\Omega_j}}$ admits a non vanishing 
holomorphic section
$$[x]\longrightarrow\varepsilon_j([x])=x/x_j=z_0e_0+\ldots
+e_j+z_{j+1}e_{j+1}+\ldots+z_ne_n,$$
and this shows in particular that $\cO(-1)$ is a holomorphic line bundle.

\claim{(15.2) Definition} For every $k\in\bZ$, the line bundle $\cO(k)$
is defined by
$$\eqalign{
\cO(1) &=\cO(-1)^\star,~~~~\cO(0)=P(V)\times\bC,\cr
\cO(k) &= \cO(1)^{\otimes k} = \cO(1) \otimes\cdots\otimes \cO(1)
\quad\hbox{\rm for}\quad k\ge 1,\cr
\cO(-k) &=\cO(-1)^{\otimes k}\quad\hbox{\rm for}\quad k\ge 1\cr}$$
\endclaim

We also introduce the quotient vector bundle $H=\soul V/\cO(-1)$ of rank~$n$.
Therefore we have canonical exact sequences of vector bundles over $P(V)\,$:
$$0\to\cO(-1)\to\soul V\to H\to 0,~~~~0\to H^\star\to\soul V^\star\to\cO(1)\to 0.
\leqno(15.3)$$
\medskip
The total manifold of the line bundle $\cO(-1)$ gives rise to the so
called {\it monoidal transformation}, or {\it Hopf $\sigma$-process}\/:

\claim{(15.4) Lemma} The holomorphic map $\mu: \cO(-1) \to V$ defined by
$$\mu: \cO(-1) \lhra \soul V= P(V)\times V\buildo pr_2\over\longrightarrow V$$
sends the zero section $P(V)\times\{0\}$ of $\cO(-1)$ to the point $\{0\}$ and 
induces a biholomorphism of $\cO(-1)\ssm \big(P(V)\times\{0\}\big)$ onto 
$V\ssm\{0\}$.
\endclaim

\proof{Proof} The inverse map $\mu^{-1}: V\ssm\{0\}\longrightarrow \cO(-1)$ is 
clearly defined by
$$\mu^{-1}: x \longmapsto \big([x],x\big).\eqno{\square}$$
\endproof

The space $H^0(\bP^n,\cO(k))$ of global holomorphic sections of $\cO(k)$
can be easily computed by means of the above map $\mu$.

\claim{(15.5) Theorem} $H^0\big(P(V),\cO(k)\big) = 0$ for $k<0$, and there
is a canonical isomorphism
$$H^0\big(P(V),\cO(k)\big) \simeq S^kV^\star,~~~~k\ge 0,$$
where $S^kV^\star$ denotes the $k$-th symmetric power of  $V^\star$.
\endclaim

\claim{(15.6) Corollary} We have $\dim H^0\big(\bP^n,\cO(k)\big)={n+k\choose n}$
for $k\ge 0$, and this group is $0$ for $k<0.$
\endclaim

\proof{Proof} Assume first that $k\ge 0$. There exists a canonical morphism 
$$\Phi: S^kV^\star\longrightarrow H^0\big(P(V),\cO(k)\big)\,;$$
indeed, any element $a\in S^kV^\star$ defines a homogeneous polynomial of 
degree $k$ on $V$ and thus by restriction to $\cO(-1)\subset\soul V$ a section 
$\Phi(a)=\wt a$ of $(\cO(-1)^\star)^{\otimes k}=\cO(k)\,$; in other words
$\Phi$ is induced by the $k$-th symmetric power $S^k\soul V^\star\to\cO(k)$
of the canonical morphism $\soul V^\star\to\cO(1)$ in (15.3).

Assume now that $k\in \bZ$ is arbitrary and that $s$ is a holomorphic 
section of $\cO(k)$. For every $x\in V\ssm\{0\}$ we have
$s([x])\in\cO(k)_{[x]}$ and \hbox{$\mu^{-1}(x)\in\cO(-1)_{[x]}$.}
We can therefore associate to $s$ a holomorphic function on $V\ssm
\{0\}$ defined by
$$f(x) = s([x])\cdot \mu^{-1}(x)^k,~~~~x\in V\ssm\{0\}.$$
Since $\dim V=n+1\ge 2$, $f$ can be extended to a holomorphic function on 
$V$ and $f$ is clearly homogeneous of degree $k$ ($\mu$ and $\mu^{-1}$ are
homogeneous of degree $1$). It follows that $f=0$,
$s=0$ if $k<0$ and that $f$ is a homogeneous polynomial of degree $k$ on 
$V$ if $k\ge 0$. Thus, there exists a unique element $a\in S^kV^\star$ 
such that
$$f(x) = a\cdot x^k = \wt a([x])\cdot\mu^{-1}(x)^k.$$
Therefore  $\Phi$ is an isomorphism.\qed
\endproof

The tangent bundle on $\bP^n$ is closely related to the bundles $H$ and
$\cO(1)$ as shown by the following proposition.

\claim{(15.7) Proposition} There is a canonical isomorphism of bundles
$$TP(V)\simeq H\otimes\cO(1).$$
\endclaim

\proof{Proof} The differential $d\pi_x$ of the projection
$\pi:V\ssm\{0\}\to P(V)$ may be considered as a map
$$d\pi_x: V\to T_{[x]} P(V).$$
As $d\pi_x(x) = 0,~d\pi_x$ can be factorized through 
$V/\bC\cdot x=V/\cO(-1)_{[x]}=H_{[x]}.$ Hence we get an isomorphism
$$d\wt \pi_x: H_{[x]}\longrightarrow T_{[x]}P(V),$$
but this isomorphism depends on $x$ and not only on the base point 
$[x]$ in~$P(V)$. The formula $\pi(\lambda x+\xi)=\pi(x+\lambda^{-1}\xi),~
\lambda\in\bC^\star,~\xi\in V$, shows that $d\pi_{\lambda x} = 
\lambda^{-1}d\pi_x$, hence the map
$$d\wt\pi_x\otimes\mu^{-1}(x)~:~~~H_{[x]}\longrightarrow\big(TP(V)\otimes\cO(-1)\big)_{[x]}$$
depends only on $[x]$. Therefore $H\simeq TP(V)\otimes\cO(-1)$.\qed
\endproof

\subsection{15.B}{Curvature of the Tautological Line Bundle}
Assume now that $V$ is a {\it hermitian vector space}. Then (15.3) yields
exact sequences of hermitian vector bundles. We shall compute the curvature of
$\cO(1)$ and $H$. 

Let $a\in P(V)$ be fixed. Choose an orthonormal basis 
$(e_0,e_1,\ldots,e_n)$ of $V$ such that $a=[e_0]$. Consider the embedding
$$\bC^n \lhra P(V),~~~~0\longmapsto a$$
which sends $z = (z_1,\ldots,z_n)$ to $[e_0+z_1e_1+\cdots+z_ne_n]$. Then
$$\varepsilon(z)=e_0+z_1e_1+\cdots+z_ne_n$$
defines a non-zero holomorphic
section of $\cO(-1)_{\restriction\bC^n}$ and Formula (13.3) for
$\Theta\big(\cO(1)\big)=-\Theta\big(\cO(-1)\big)$ implies
$$\leqalignno{
~~~~~~~~\Theta\big(\cO(1)\big)&= d'd''\log|\varepsilon(z)|^2=d'd'' \log(1+|z|^2)
~~~~\hbox{\rm on}~\bC^n,&(15.8)\cr
\Theta\big(\cO(1)\big)_a&=\sum_{1\le j\le n}dz_j\wedge d\overline z_j.
&(15.8')\cr}$$
On the other hand, Th.~14.3 and (14.7) imply
$$d''g^\star=-j\circ\beta^\star,~~~~\Theta(H)=\beta\wedge\beta^\star,$$
where 
$j:\cO(-1)\longrightarrow\soul V$ is the inclusion, $g^\star:H\longrightarrow\soul V$ the orthogonal splitting
and $\beta^\star\in \cC^\infty_{0,1}\big(P(V),\Hom(H,\cO(-1))\big)$. The
images $(\wt e_1,\ldots,\wt e_n)$ of $e_1,\ldots,e_n$ in $H=\soul V/\cO(-1)$ define a
holomorphic frame of $H_{\restriction\bC^n}$ and we have
$$\leqalignno{
g^\star\cdot\wt e_j&=e_j-{\langle e_j,\varepsilon\rangle\over|\varepsilon|^2}
=e_j-{\ol z_j\over 1+|z|^2}\varepsilon,~~~~d''g_a^\star\cdot\wt e_j=-d\ol z_j
\otimes\varepsilon,\cr
\beta^\star_a&=\sum_{1\le j\le n}d\overline z_j\otimes\wt e_j^\star
\otimes\varepsilon,~~~~
\beta_a=\sum_{1\le j\le n}dz_j\otimes\varepsilon^\star\otimes\wt e_j,\cr
~~~~~~~~\Theta(H)_a&=\sum_{1\le j,k\le n} dz_j\wedge d\overline z_k\otimes
\wt e_k^\star\otimes\wt e_j.&(15.9)\cr}$$

\claim{(15.10) Theorem} The cohomology algebra $H^\bu(\bP^n,\bZ)$ is isomorphic
to the quotient ring $\bZ[h]/(h^{n+1})$ where the generator $h$ is given by
$h=c_1(\cO(1))$ in $H^2(\bP^n,\bZ).$ 
\endclaim

\proof{Proof} Consider the inclusion $\bP^{n-1}=P(\bC^n\times\{0\})
\subset\bP^n.$ Topologically, $\bP^n$~is obtained from $\bP^{n-1}$ by
attaching a $2n$-cell $B_{2n}$ to $\bP^{n-1}$, via the map
$$\eqalign{
f~:~~B_{2n}&\longrightarrow\bP^n\cr
z&\longmapsto[z,1-|z|^2],~~~~z\in\bC^n,~~|z|\le 1\cr}$$
which sends $S^{2n-1}=\{|z|=1\}$ onto $\bP^{n-1}$.
That is, $\bP^n$ is homeomorphic to the quotient space of $B_{2n}\amalg
\bP^{n-1}$, where every point $z\in S^{2n-1}$ is identified with its
image $f(z)\in\bP^{n-1}$. We shall prove by induction on $n$ that
$$H^{2k}(\bP^n,\bZ)=\bZ,~~0\le k\le n,~~
\hbox{\rm otherwise}~~H^l(\bP^n,\bZ)=0.\leqno(15.11)$$
The result is clear for $\bP^0$, which is reduced to a single point.
For $n\ge 1$, consider the covering $(U_1,U_2)$ of $\bP^n$ such that
$U_1$ is the image by $f$ of the open ball $B_{2n}^\circ$ and 
$U_2=\bP^n\ssm\{f(0)\}$. Then $U_1\approx B_{2n}^\circ$ is
contractible, whereas $U_2=(B_{2n}\ssm\{0\})\amalg_{S^{2n-1}}\bP^{n-1}$.
Moreover $U_1\cap U_2
\approx B_{2n}^\circ\ssm\{0\}$ can be retracted on the $(2n-1)$-sphere
of radius $1/2$. For $q\ge 2$, the Mayer-Vietoris exact sequence IV-3.11 
yields
$$\eqalign{
\cdots~~H^{q-1}(\bP^{n-1},\bZ)&\longrightarrow H^{q-1}(S^{2n-1},\bZ)\cr
\longrightarrow H^q(\bP^n,\bZ)\longrightarrow H^q(\bP^{n-1},\bZ)&\longrightarrow H^q(S^{2n-1},\bZ)
~~\cdots~.\cr}$$
For $q=1$, the first term has to be replaced by $H^{0}(\bP^{n-1},\bZ)\oplus
\bZ$, so that the first arrow is onto. Formula (15.11) follows
easily by induction, thanks to our computation of the cohomology groups of 
spheres in IV-14.6.

We know that $h=c_1(\cO(1))\in H^2(\bP^n,\bZ)$. It will follow necessarily
that $h^k$ is a generator of $H^{2k}(\bP^n,\bZ)$ if we can prove that
$h^n$ is the fundamental class in $H^{2n}(\bP,\bZ)$, or
equivalently that
$$c_1\big(\cO(1)\big)_\bR^n=\int_{\bP^n}\Big({\ii\over 2\pi}\Theta(\cO(1))
\Big)^n=1.\leqno(15.12)$$
This equality can be verified directly by means of (15.8), but we will avoid
this computation. Observe that the element 
$e_n^\star\in\big(\bC^{n+1}\big)^\star$ defines 
a section ${\wt e_n}^\star$ of $H^0(\bP^n,\cO(1))$ transverse to 0, whose 
zero set is the hyperplane $\bP^{n-1}$. As $\{{\ii\over 2\pi}\Theta(\cO(1))\}=
\{[\bP^{n-1}]\}$ by Th.~13.4, we get
$$\eqalign{
c_1(\cO(1))&=\int_{\bP^1}[\bP^0]=1~~~~\hbox{\rm for}~~n=1~~\hbox{\rm and}\cr
c_1(\cO(1))^n&=\int_{\bP^n}[\bP^{n-1}]\wedge\Big({\ii\over 2\pi}
\Theta(\cO(1))\Big)^{n-1}=
\int_{\bP^{n-1}}\Big({\ii\over 2\pi}\Theta(\cO(1))\Big)^{n-1}\cr}$$
in general. Since $\cO(-1)_{\restriction\bP^{n-1}}$ can be identified with
the tautological line subbundle $\cO_{\bP^{n-1}}(-1)$ over
$\bP^{n-1}$, we have $\Theta(\cO(1))_{\restriction\bP^{n-1}}=
\Theta(\cO_{\bP^{n-1}}(1))$ and the proof is achieved by induction
on $n$.\qed
\endproof

\subsection{15.C}{Tautological Line Bundle Associated to a Vector Bundle}
Let $E$ be a holomorphic vector bundle of rank $r$ over a complex
manifold~$X$. The projectivized bundle $P(E)$ is the bundle with
$\bP^{r-1}$ fibers over $X$ defined by $P(E)_x=P(E_x)$ for all $x\in X$.
The points of $P(E)$ can thus be identified with the lines in the fibers
of~$E$. For any trivialization \hbox{$\theta_\alpha:E_{\restriction U_\alpha}
\to U_\alpha\times\bC^r$} of $E$ we have a corresponding trivialization
\hbox{$\wt\theta_\alpha:P(E)_{\restriction U_\alpha}\to U_\alpha\times
\bP^{r-1}$,} and it is clear that the transition automorphisms are the
projectivizations
\hbox{$\wt g_{\alpha\beta}\in H^0\big(U_\alpha\cap U_\beta,PGL(r,\bC)\big)$}
of the transition automorphisms $g_{\alpha\beta}$ of~$E$.

Similarly, we have a dual projectivized bundle $P(E^\star)$ whose
points can be identified with the hyperplanes of $E$ (every hyperplane
$F$ in $E_x$ corresponds bijectively to the line of linear forms in
$E^\star_x$ which vanish on $F$); note that $P(E)$ and $P(E^\star)$
coincide only when $r=\rk E=2$. If $\pi:P(E^\star)\to X$ is the natural
projection, there is a tautological hyperplane subbundle $S$ of
$\pi^\star E$ over $P(E^\star)$ such that $S_{[\xi]}=\xi^{-1}(0)\subset
E_x$ for all $\xi\in E^\star_x\ssm\{0\}$.\Fbreak
$\big[$exercise: check that $S$ is
actually locally trivial over $P(E^\star)\big]$.

\claim{(15.13) Definition} The quotient line bundle $\pi^\star E/S$ is denoted
$\cO_E(1)$ and is called the tautological line bundle associated to~$E$.
Hence there is an exact sequence
$$0\longrightarrow S\longrightarrow\pi^\star E\longrightarrow\cO_E(1)\longrightarrow 0$$
of vector bundles over $P(E^\star)$.
\endclaim

Note that (13.3) applied with $V=E^\star_x$ implies that the restriction
of $\cO_E(1)$ to each fiber $P(E^\star_x)\simeq\bP^{r-1}$ coincides
with the line bundle $\cO(1)$ introduced in Def.~15.2. Theorem~15.5
can then be extended to the present situation and yields:

\claim{(15.14) Theorem} For every $k\in\bZ$, the direct image sheaf
$\pi_\star\cO_E(k)$ on $X$ vanishes for $k<0$ and is isomorphic
to $\cO(S^kE)$ for $k\ge 0$.
\endclaim

\proof{Proof} For $k\ge 0$, the $k$-th symmetric power of the morphism
$\pi^\star E\to\cO_E(1)$ gives a morphism $\pi^\star S^kE\to\cO_E(k)$.
This morphism together with the pull-back morphism yield canonical arrows
$$\Phi_U:H^0(U,S^kE)\buildo\pi^\star\over\longrightarrow
H^0\big(\pi^{-1}(U),\pi^\star S^kE\big)\longrightarrow
H^0\big(\pi^{-1}(U),\cO_E(k)\big)$$
for any open set $U\subset X$. The right hand side is by definition the
space of sections of $\pi_\star\cO_E(k)$ over $U$, hence we get a canonical
sheaf morphism
$$\Phi:\cO(S^kE)\longrightarrow\pi_\star\cO_E(k).$$
It is easy to check that this $\Phi$ coincides with the map $\Phi$
introduced  in the proof of Cor.~15.6 when $X$ is reduced to a point.
In order to check that $\Phi$ is an isomorphism, we may suppose that
$U$ is chosen so small that $E_{\restriction U}$ is trivial, say
$E_{\restriction U}=U\times V$ with $\dim V=r$. Then
$P(E^\star)=U\times P(V^\star)$ and \hbox{$\cO_E(1)=p^\star\cO(1)$} where
$\cO(1)$ is the tautological line bundle over $P(V^\star)$ and
\hbox{$p:P(E^\star)\to P(V^\star)$} is the second projection. Hence we get
$$\eqalign{
H^0\big(\pi^{-1}(U),\cO_E(k)\big)
&=H^0\big(U\times P(V^\star),p^\star\cO(1)\big)\cr
&=\cO_X(U)\otimes H^0\big(P(V^\star),\cO(1)\big)\cr
&=\cO_X(U)\otimes S^kV=H^0(U,S^kE),\cr}$$
as desired; the reason for the second equality is that $p^\star\cO(1)$
coincides with $\cO(1)$ on each fiber $\{x\}\times P(V^\star)$ of~$p$,
thus any section of $p^\star\cO(1)$ over $U\times P(V^\star)$ yields a
family of sections $H^0\big(\{x\}\times P(V^\star),\cO(k)\big)$
depending holomorphically in~$x$. When $k<0$ there are no non zero such
sections, thus $\pi_\star\cO_E(k)=0$.\qed
\endproof

Finally, suppose that $E$ is equipped with a hermitian metric. Then
the morphism $\pi^\star E\to\cO_E(1)$ endows $\cO_E(1)$ with a
quotient metric. We are going to compute the associated curvature
form~$\Theta\big(\cO_E(1)\big)$.

Fix a point $x_0\in X$ and $a\in P(E^\star_{x_0})$. Then Prop.~12.10
implies the existence of a normal coordinate frame $(e_\lambda)_{1\le
\lambda\le r})$ of $E$ at $x_0$ such that $a$ is the hyperplane
$\langle e_2,\ldots,e_r\rangle=(e^\star_1)^{-1}(0)$ at~$x_0$. Let
$(z_1,\ldots,z_n)$ be local coordinates on $X$ near~$x_0$ and let
$(\xi_1,\ldots,\xi_r)$ be coordinates on $E^\star$ with respect to the dual
frame~$(e^\star_1,\ldots,e^\star_r)$. If we assign $\xi_1=1$, then
$(z_1,\ldots,z_n,\xi_2,\ldots,\xi_r)$ define local coordinates on $P(E^\star)$
near~$a$, and we have a local section of $\cO_E(-1):=
\cO_E(1)^\star\subset\pi^\star E^\star$ defined by
$$\varepsilon(z,\xi)=e^\star_1(z)+\sum_{2\le\lambda\le r}\xi_\lambda
\,e^\star_\lambda(z).$$
The hermitian matrix $(\langle e^\star_\lambda,e^\star_\mu\rangle)$ is
just the congugate of the inverse of matrix 
$(\langle e_\lambda,e_\mu\rangle)=\Id
-\big(\sum c_{jk\lambda\mu}\,z_j\ol z_k\big)+O(|z|^3)$, hence we get
$$\langle e^\star_\lambda(z),e^\star_\mu(z)\rangle=\delta_{\lambda\mu}+
\sum_{1\le j,k\le n}c_{jk\mu\lambda}\,z_j\ol z_k+O(|z|^3),$$
where $(c_{jk\lambda\mu})$ are the curvature coefficients of $\Theta(E)\,$;
accordingly we have $\Theta(E^\star)=-\Theta(E)^\dagger$. We infer from this
$$|\varepsilon(z,\xi)|^2=1+\sum_{1\le j,k\le n}c_{jk11}\,z_j\ol z_k+
\sum_{2\le\lambda\le r}|\xi_\lambda|^2+O(|z|^3).$$
Since $\Theta\big(\cO_E(1)\big)=d'd''\log|\varepsilon(z,\xi)|^2$, we get
$$\Theta\big(\cO_E(1)\big)_a=\sum_{1\le j,k\le n}c_{jk11}\,dz_j\wedge d\ol z_k
+\sum_{2\le\lambda\le r}d\xi_\lambda\wedge d\ol\xi_\lambda.$$
Note that the first summation is simply $-\langle\Theta(E^\star)a,a\rangle/
|a|^2=-{}$ curvature of $E^\star$ in the direction~$a$.
A unitary change of variables then gives the slightly more general formula:

\claim{(15.15) Formula} Let $(e_\lambda)$ be a normal coordinate
frame of $E$ at~$x_0\in X$ and let $\Theta(E)_{x_0}=\sum
c_{jk\lambda\mu}\,dz_j\wedge d\ol z_k\otimes e_\lambda^\star\otimes e_\mu$.
At any point $a\in P(E^\star)$ represented
by a vector $\sum a_\lambda e^\star_\lambda\in E^\star_{x_0}$ of norm~$1$,
the curvature of $\cO_E(1)$ is
$$\Theta\big(\cO_E(1)\big)_a=\sum_{1\le j,k\le n,\,1\le\lambda,\mu\le r}
c_{jk\mu\lambda}\,a_\lambda\ol a_\mu\,dz_j\wedge d\ol z_k
+\sum_{1\le\lambda\le r-1}d\zeta_\lambda\wedge d\ol\zeta_\lambda,$$
where $(\zeta_\lambda)$ are coordinates near $a$ on $P(E^\star)$, induced
by unitary coordinates on the hyperplane $a^\perp\subset E^\star_{x_0}$.\qed
\endclaim

\section{16}{Grassmannians and Universal Vector Bundles}
\subsection{16.A}{Universal Subbundles and Quotient Vector Bundles}
If $V$ is a complex vector space of dimension $d$, we denote by
$G_r(V)$ the set of all $r$-codimensional vector subspaces of $V$. Let
$a\in G_r(V)$ and $W\subset V$ be fixed such that
$$V=a\oplus W,~~~~\dim_\bC W=r.$$
Then any subspace $x\in G_r(V)$ in the open subset
$$\Omega_W=\{x\in G_r(V)~;~x\oplus W=V\}$$
can be represented in a unique way as the graph of a linear map $u$ in
$\Hom(a,W)$. This gives rise to a covering of $G_r(V)$ by affine
coordinate charts $\Omega_W\simeq\Hom(a,W)\simeq\bC^{r(d-r)}$.
Indeed, let $(e_1,\ldots,e_r)$ and $(e_{r+1},\ldots,e_n)$ be respective bases
of $W$ and $a$. Every point $x\in\Omega_W$ is the graph of a linear map
$$u:~a\longrightarrow W,~~~~u(e_k)=\sum_{1\le j\le r}z_{jk}e_j, ~~~
r+1\le k\le d,\leqno(16.1)$$
i.e.\ $x=\Vect\big(e_k+\sum_{1\le j\le r}z_{jk}e_j\big)_{r+1\le k\le d}$.
We choose $(z_{jk})$ as complex coordinates on $\Omega_W$. These
coordinates are centered at $a=\Vect(e_{r+1},\ldots,e_d)$.

\claim{(16.2) Proposition} $G_r(V)$ is a compact complex analytic 
manifold of dimension $n=r(d-r)$.
\endclaim

\proof{Proof} It is immediate to verify that the coordinate change between
two affine charts of $G_r(V)$ is holomorphic. Fix an arbitrary hermitian 
metric~on~$V\,$. Then the unitary group $U(V)$ is compact and acts 
transitively on $G_r(V)$. The isotropy subgroup of a point $a\in G_r(V)$ is 
$U(a)\times U(a^\perp)$, hence $G_r(V)$ is diffeomorphic to the compact 
quotient space $U(V)/U(a)\times U(a^\perp)$.\qed
\endproof

Next, we consider the tautological subbundle $S\subset\soul V:=G_r(V)\times V$
defined by $S_x=x$ for all $x\in G_r(V)$, and the quotient bundle
$Q=\soul V/S$ of rank $r\,$:
$$0\longrightarrow S\longrightarrow\soul V\longrightarrow Q\longrightarrow 0.\leqno(16.3)$$
An interesting special case is $r=d-1$, $G_{d-1}(V)=P(V)$, $S=\cO(-1)$, 
$Q=H$. The case $r=1$ is dual, we have the identification $G_1(V)=P(V^\star)$
because every hyperplane $x\subset V$ corresponds bijectively to the line in
$V^\star$ of linear forms $\xi\in V^\star$ that vanish on $x$. Then the
bundles $\cO(-1)\subset\soul V^\star$ and $H$ on $P(V^\star)$ are given by 
$$\eqalign{
\cO(-1)_{[\xi]}&=\bC.\xi\simeq(V/x)^\star=Q_x^\star,\cr
H_{[\xi]}&=V^\star/\bC.\xi\simeq x^\star=S_x^\star,\cr}$$
therefore $S=H^\star$, $Q=\cO(1)$. This special case will allow us to
compute $H^0(G_r(V),Q)$ in general.

\claim{(16.4) Proposition} There is an isomorphism
$$V=H^0\big(G_r(V),\soul V\big)\buildo\sim\over\longrightarrow H^0\big(G_r(V),Q\big).$$
\endclaim

\proof{Proof} Let $V=W\oplus W'$ be an arbitrary direct sum decomposition of
$V$ with $\codim W=r-1$. Consider the projective space
$$P(W^\star)=G_1(W)\subset G_r(V),$$
its tautological hyperplane subbundle $H^\star\subset\soul W=P(W^\star)
\times W$ and the exact sequence $0\to H^\star\to\soul W\to\cO(1)\to 0$.
Then $S_{\restriction P(W^\star)}$ coincides with $H^\star$ and
$$Q_{\restriction P(W^\star)}=(\soul W\oplus\soul W')/H^\star
=(\soul W/H^\star)\oplus\soul W'=\cO(1)\oplus\soul W'.$$
Theorem~15.5 implies $H^0(P(W^\star),\cO(1))=W$, therefore the space
$$H^0(P(W^\star),Q_{\restriction P(W^\star)})=W\oplus W'$$
is generated by the images of the 
constant sections of $\soul V$. Since $W$ is arbitrary,
Prop.~16.4 follows immediately.\qed
\endproof

Let us compute the tangent space $TG_r(V)$. The linear group $\Gl(V)$ 
acts transitively on $G_r(V)$, and the tangent
space to the isotropy subgroup of a point $x\in G_r(V)$ is the set of
elements $u\in \Hom(V,V)$ in the Lie algebra such that $u(x)\subset x$.
We get therefore
$$\eqalign{T_xG_r(V)&\simeq \Hom(V,V)/\{u~;~u(x)\subset x\}\cr
                    &\simeq \Hom(V,V/x)/\big\{\wt u~;~\wt u(x)=\{0\}\big\}\cr
                    &\simeq \Hom(x,V/x)=\Hom(S_x,Q_x).\cr}$$

\claim{(16.5) Corollary} $TG_r(V)=\Hom(S,Q)=S^\star\otimes Q$.\qed
\endclaim

\subsection{16.B}{Pl\"ucker Embedding}
There is a natural map, called the {\it Pl\"ucker embedding},
$$j_r:G_r(V)\lhra P(\Lambda^rV^\star)\leqno(16.6)$$
constructed as follows. If $x\in G_r(V)$ is defined by $r$ independent linear 
forms $\xi_1,\ldots,\xi_r\in V^\star$, we set
$$j_r(x)=[\xi_1\wedge\cdots\wedge\xi_r].$$
Then $x$ is the subspace of vectors $v\in V$ such that
$v\ort(\xi_1\wedge\cdots\wedge\xi_r)=0$, so $j_r$ is injective. Since
the linear group $\Gl(V)$ acts transitively on $G_r(V)$, the
rank of the differential $dj_r$ is a constant. As $j_r$ is injective,
the constant rank theorem implies:

\claim{(16.7) Proposition} The map $j_r$ is a holomorphic embedding.\qed
\endclaim

Now, we define a commutative diagram
$$\matrix{
\Lambda^rQ&\buildo{\displaystyle J_r}\over\longrightarrow&\cO(1)\cr
\downarrow&&\downarrow\cr
G_r(V)&\buildo{\displaystyle j_r}\over \lhra&P(\Lambda^rV^\star)\cr}
\leqno(16.8)$$
as follows: for $x=\xi_1^{-1}(0)\cap\cdots\cap\xi_r^{-1}(0)\in 
G_r(V)$ and \hbox{$\wt v=\wt v_1\wedge\cdots\wedge\wt v_r\in\Lambda^r Q_x$}
where $\wt v_k\in Q_x=V/x$ is the image of $v_k\in V$ in the
quotient, we let \hbox{$J_r(\wt v)\in\cO(1)_{j_r(x)}$} be the linear form
on $\cO(-1)_{j_r(x)}=\bC.\xi_1\wedge\ldots\wedge\xi_r$ such that
$$\langle J_r(\wt v),\lambda\xi_1\wedge\ldots\wedge\xi_r\rangle=
\lambda\det\big(\xi_j(v_k)\big),~~~~\lambda\in\bC.$$
Then $J_r$ is an isomorphism on the fibers, so $\Lambda^rQ$ can be 
identified with the pull-back of $\cO(1)$ by $j_r$.

\subsection{16.C}{Curvature of the Universal Vector Bundles}
Assume now that $V$ is a hermitian vector space. We shall generalize our
curvature computations of \S 15.C to the present situation. Let $a\in G_r(V)$
be a given point. We take $W$ to be the orthogonal complement of $a$ 
in $V$ and select an orthonormal basis $(e_1,\ldots,e_d)$ of $V$ such
that $W=\Vect(e_1,\ldots,e_r)$, $a=\Vect(e_{r+1},\ldots,e_d)$. For any
point $x\in G_r(V)$ in $\Omega_W$ with coordinates $(z_{jk})$, we set
$$\eqalign{
\varepsilon_k(x)&=e_k+\sum_{1\le j\le r}z_{jk}e_j,~~~~r+1\le k\le d,\cr
\wt e_j(x)&=\hbox{\rm image of}~e_j~\hbox{\rm in}~Q_x=V/x,~~~~1\le j\le r.\cr}$$
Then $(\wt e_1,\ldots,\wt e_r)$ and $(\varepsilon_{r+1},\ldots,\varepsilon_d)$ are
holomorphic frames of $Q$ and $S$ respectively. If $g^\star:~Q\longrightarrow\soul V$ 
is the orthogonal splitting of $g:\soul V\longrightarrow Q$, then
$$g^\star\cdot\wt e_j=e_j+\sum_{r+1\le k\le d}\zeta_{jk}\varepsilon_k$$
for some $\zeta_{jk}\in\bC$. After an easy computation we find
$$0=\langle\wt e_j,g\varepsilon_k\rangle=\langle g^\star\wt e_j,\varepsilon_k
\rangle=\zeta_{jk}+\ol z_{jk}+\sum_{l,m}\zeta_{jm}z_{lm}\ol z_{lk},$$
so that $\zeta_{jk}=-\ol z_{jk}+O(|z|^2)$. Formula (13.3) yields
$$\leqalignno{
d''g^\star_a\cdot\wt e_j&=-\sum_{r+1\le k\le d}d\ol z_{jk}\otimes\varepsilon_k,\cr
\beta^\star_a&=\sum_{j,k}d\ol z_{jk}\otimes\wt e_j^\star\otimes\varepsilon_k,
~~~~\beta_a=\sum_{j,k}dz_{jk}\otimes\varepsilon_k^\star\otimes\wt e_j,\cr
\Theta(Q)_a&=(\beta\wedge\beta^\star)_a=\sum_{j,k,l}
dz_{jk}\wedge d\ol z_{lk}\otimes\wt e_l^\star\otimes\wt e_j,&(16.9)\cr
\Theta(S)_a&=(\beta^\star\wedge\beta)_a=-\sum_{j,k,l}dz_{jk}\wedge d\ol z_{jl}\otimes
\varepsilon^\star_k\otimes\varepsilon_l.&(16.10)\cr}$$
 

\chapter{Chapter VI}{Hodge Theory}
\begineightpoint
The goal of this chapter is to prove a number of basic facts in the Hodge 
theory of real or complex manifolds. The theory rests essentially on the fact 
that the De Rham (or Dolbeault) cohomology groups of a compact manifold can 
be represented by means of spaces of harmonic forms, once a Riemannian 
metric has been chosen. At this point, some knowledge of basic
results about elliptic differential operators is required. The special
properties of compact K\"ahler manifolds are then investigated in detail: 
Hodge decomposition theorem, hard Lefschetz theorem, Jacobian and Albanese
variety, $\ldots\,$; the example of curves is treated in detail. Finally,
the Hodge-Fr\"olicher spectral sequence is applied to get some 
results on general compact complex manifolds, and it is shown that
Hodge decomposition still holds for manifolds in the Fujiki class
$(\cC)$.
\endeightpoint

\section{1}{Differential Operators on Vector Bundles}
We first describe some basic concepts concerning differential operators
(symbol, composition, adjunction, ellipticity), in the general setting
of vector bundles. Let $M$ be a $\cC^\infty$ differentiable manifold,
$\dim_{\bR}M=m$, and let $E$, $F$ be $\bK$-vector bundles over $M$,
with $\bK=\bR$ or $\bK=\bC$, $\rank E=r$, $\rank F=r'$.

\claim{(1.1) Definition} A $($linear$)$ differential operator of degree
$\delta$ from $E$ to $F$ is a $\bK$-linear operator
$P:\cC^\infty(M,E)\to \cC^\infty(M,F)$, $u\mapsto Pu$ of the form
$$Pu(x)=\sum_{|\alpha|\le\delta}a_\alpha(x)D^\alpha u(x),$$
where $E_{\restriction\Omega}\simeq\Omega\times\bK^r$,
$F_{\restriction\Omega}\simeq\Omega\times\bK^{r'}$ are trivialized
locally on some open chart \hbox{$\Omega\subset M$} equipped with local
coordinates $(x_1,\ldots,x_m)$, and where
\hbox{$a_\alpha(x)=\big(a_{\alpha\lambda\mu}
(x)\big)_{1\le\lambda\le r',\,1\le\mu\le r}$} are $r'\times r$-matrices
with $\cC^\infty$ coefficients on $\Omega$. Here \hbox{$D^\alpha=(\partial/
\partial x_1)^{\alpha_1}\ldots(\partial/\partial x_m)^{\alpha_m}$}
as usual, and \hbox{$u=(u_\mu)_{1\le\mu\le r}$},
\hbox{$D^\alpha u=(D^\alpha u_\mu)_{1\le\mu\le r}$} are viewed as column
matrices.
\endclaim

If $t\in\bK$ is a parameter and $f\in C^\infty(M,\bK)$, $u\in C^\infty(M,E)$, 
a simple calculation shows that
$e^{-tf(x)}P(e^{tf(x)}u(x))$ is a polynomial of degree $\delta$ in $t$, of
the form
$$e^{-tf(x)}P(e^{tf(x)})=t^\delta\sigma_P(x,df(x))\cdot u(x)+
\hbox{\rm lower order terms $c_j(x)t^j$, $j<\delta$},$$
where $\sigma_P$ is the polynomial map from $T^\star_M\to\Hom(E,F)$
defined by
$$T^\star_{M,x}\ni\xi\mapsto\sigma_P(x,\xi)\in\Hom(E_x,F_x),\qquad
\sigma_P(x,\xi)=\sum_{|\alpha|=\delta}a_\alpha(x)\xi^\alpha.\leqno(1.2)$$
The formula involving $e^{-tf}P(e^{tf}u)$ shows that $\sigma_P(x,\xi)$
actually does not depend on the choice of coordinates nor on the
trivializations used for $E$, $F$. It is clear that $\sigma_P(x,\xi)$ is
smooth on $T^\star_M$ as a function of $(x,\xi)$, and is a homogeneous
polynomial of degree $\delta$ in~$\xi$. We say that $\sigma_P$ is {\it the
principal symbol} of~$P$. Now, if $E$, $F$, $G$ are vector bundles and
$$P:\cC^\infty(M,E)\to \cC^\infty(M,F),\qquad
Q:\cC^\infty(M,F)\to \cC^\infty(M,G)$$
are differential operators of respective degrees $\delta_P$, $\delta_Q$, it
is easy to check that $Q\circ P:\cC^\infty(M,E)\to \cC^\infty(M,G)$ is a
differential operator of degree $\delta_P+\delta_Q$ and that
$$\sigma_{Q\circ P}(x,\xi)=\sigma_{Q}(x,\xi)\sigma_{P}(x,\xi).\leqno(1.3)$$
Here the product of symbols is computed as a product of matrices.

Now, assume that $M$ is oriented and is equipped with a smooth volume
form $dV(x)=\gamma(x)dx_1\wedge\ldots dx_m$, where $\gamma(x)>0$ is a
smooth density. If $E$ is a euclidean or hermitian vector bundle,
we have a Hilbert space $L^2(M,E)$ of global sections $u$ of~$E$ with
measurable coefficients, satisfying the $L^2$ estimate
$$\|u\|^2=\int_M|u(x)|^2\,dV(x)<+\infty.\leqno(1.4)$$
We denote by
$$\Ll u,v\Gg=\int_M\langle u(x),v(x)\rangle\,dV(x),\qquad
u,v\in L^2(M,E)\leqno(1.4')$$
the corresponding $L^2$ inner product.

\claim{(1.5) Definition} If $P:\cC^\infty(M,E)\to \cC^\infty(M,F)$
is a differential operator and both $E$, $F$ are euclidean or hermitian,
there exists a unique differential operator
$$P^\star:\cC^\infty(M,F)\to \cC^\infty(M,E),$$
called the {\it formal adjoint} of $P$, such that for all sections
$u\in \cC^\infty(M,E)$ and $v\in \cC^\infty(M,F)$ there is an identity
$$\Ll Pu,v\Gg=\Ll u,P^\star v\Gg,\qquad\hbox{\rm whenever
$\Supp u\cap\Supp v\compact M$}.$$
\endclaim

\proof{Proof} The uniqueness is easy, using the density of the set
of elements $u\in \cC^\infty(M,E)$ with compact support in $L^2(M,E)$.
Since uniqueness is clear, it is enough, by a partition of unity argument,
to show the existence of $P^\star$ locally. Now, let
$Pu(x)=\sum_{|\alpha|\le\delta}a_\alpha(x)D^\alpha u(x)$
be the expansion of $P$ with respect to trivializations of $E$, $F$ given
by orthonormal frames over some coordinate open set $\Omega\subset M$. 
Assuming
$\Supp u\cap\Supp v\compact\Omega$, an integration by parts
yields
$$\eqalign{
\Ll Pu,v\Gg&=\int_\Omega\sum_{|\alpha|\le\delta,\lambda,\mu}
a_{\alpha\lambda\mu}D^\alpha u_\mu(x)\ol v_\lambda(x)\,\gamma(x)\,
dx_1,\ldots,dx_m\cr
&=\int_\Omega\sum_{|\alpha|\le\delta,\lambda,\mu}(-1)^{|\alpha|}
u_\mu(x)\ol{D^\alpha(\gamma(x)\,\ol a_{\alpha\lambda\mu}v_\lambda(x)}
\,dx_1,\ldots,dx_m\cr
&=\int_\Omega\langle u,\sum_{|\alpha|\le\delta}(-1)^{|\alpha|}\gamma(x)^{-1}
D^\alpha\big(\gamma(x)\,{}^t\ol a_\alpha v(x)\big)\rangle\,dV(x).\cr}$$
Hence we see that $P^\star$ exists and is uniquely defined by
$$P^\star v(x)=\sum_{|\alpha|\le\delta}(-1)^{|\alpha|}\gamma(x)^{-1}D^\alpha
\big(\gamma(x)\,{}^t\ol a_\alpha v(x)\big).\leqno\squareatend
(1.6)$$%
\endproof

It follows immediately from (1.6) that the principal symbol of $P^\star$ is
$$\sigma_{P^\star}(x,\xi)=(-1)^\delta\sum_{|\alpha|=\delta}
{}^t\ol a_\alpha\xi^\alpha=(-1)^\delta\sigma_P(x,\xi)^\star.\leqno(1.7)$$

\claim{(1.8) Definition} A differential operator $P$ is said to be
elliptic if
$$\sigma_P(x,\xi)\in\Hom(E_x,F_x)$$
is injective for every $x\in M$ and $\xi\in T^\star_{M,x}\ssm\{0\}$.
\endclaim

\section{2}{Formalism of PseudoDifferential Operators}

We assume throughout this section that $(M,g)$ is a compact Riemannian
manifold. For any positive integer $k$ and any hermitian bundle $F\to
M$, we denote by $W^k(M,F)$ the Sobolev space of sections $s:M\to F$
whose derivatives up to order $k$ are in $L^2$. Let $\|~~\|_k$ be the
norm of the Hilbert space $W^k(M,F)$. Let $P$ be an elliptic
differential operator of order $d$ acting on $\cC^\infty(M,F)$. We need
the following basic facts of elliptic $PDE$ theory, see e.g.\
\cite{H\"ormander 1963}.

\claim{(2.1) Sobolev lemma} For
$k>l+{m\over 2}$, $W^k(M,F)\subset C^l(M,F)$.
\endclaim

\claim{(2.2) Rellich lemma} For every integer $k$, the inclusion 
$$W^{k+1}(M,F)\lhra W^k(M,F)$$ 
is a compact linear operator.
\endclaim

\claim{(2.3) G\aa rding's inequality} Let $\wt P$ be the extension of 
$P$ to sections with distribution coefficients. For any $u\in W^0 (M,F)$
such that $\wt P u\in W^k(M,F)$, then $u\in W^{k+d}(M,F)$ and
$$\|u\|_{k+d}\le C_k(\|\wt Pu\|_k+\|u\|_0),$$
where $C_k$ is a positive constant depending only on $k$.
\endclaim

\claim{(2.4) Corollary} The operator $P:\cC^\infty(M,F)\to \cC^\infty (M,F)$
has the following properties:
\smallskip
\item{\rm i)} $\ker P$ is finite dimensional.
\smallskip
\item{\rm ii)} $P\big(\cC^\infty(M,F)\big)$ is closed and of finite codimension; 
furthermore, if $P^\star$ is the formal adjoint of $P$, there is a 
decomposition
$$\cC^\infty(M,F)=P\big(\cC^\infty(M,F)\big)\oplus\ker P^\star$$
as an orthogonal direct sum in $W^0(M,F)=L^2(M,F)$.\smallskip
\endclaim

\proof{Proof} (i) G\aa rding's inequality shows that $\|u\|_{k+d}\le C_k\|u\|_0$
for any $u$ in $\ker P$. Thanks to the Sobolev  lemma, this implies that 
$\ker P$ is closed in $W^0(M,F)$. Moreover, the unit closed 
$\|~~\|_0$-ball of $\ker P$ is contained in the $\|~~\|_d$-ball of radius
$C_0$, thus compact by the Rellich lemma. Riesz' theorem implies that 
$\dim\ker P<+\infty$.
\medskip\noindent(ii) We first show that the extension
$$\wt P:W^{k+d}(M,F)\to W^k(M,F)$$
has a closed range for any $k$. For every $\varepsilon>0$, there  exists
a finite number of elements $v_1,\ldots,v_N\in W^{k+d}(M,F)$, $N=N(\varepsilon)$,
such that
$$\|u\|_0\le\varepsilon\|u\|_{k+d}+\sum^N_{j=1}|\Ll u,v_j\Gg_0|~;
\leqno(2.5)$$
indeed the set
$$K_{(v_j)}=\Big\{ u\in W^{k+d}(M,F)~ ;~\varepsilon\|u\|_{k+d}+
\sum^N_{j=1} |\Ll u,v_j\Gg_0|\le 1\Big\}$$
is relatively compact in $W^0(M,F)$ and $\bigcap_{(v_j)}\ol K_{(v_j)}=\{
0\}$. It follows that there exist elements $(v_j)$ such that 
$\ol K_{(v_j)}$ is contained in the unit ball of $W^0(M,F)$,~$QED$. 
Substitute $||u||_0$ by the upper bound (2.5) in G\aa rding's inequality;
we get
$$(1-C_k\varepsilon)\|u\|_{k+d}\le C_k\Big(\|\wt P u\|_k+\sum^N_{j=1}
|\Ll u,v_j\Gg_0|\Big).$$
Define $G=\big\{u\in W^{k+d}(M,F)~;~u\perp v_j,~1\le j\le n\}$ and choose 
$\varepsilon=1/2C_k$.  We obtain
$$\|u\|_{k+d}\le 2C_k\|\wt P u\|_k,~~\forall u\in G.$$
This implies that $\wt P(G)$ is closed. Therefore
$$\wt P\big(W^{k+d}(M,F)\big)=\wt P(G)+{\rm Vect}
\big(\wt P(v_1),\ldots,\wt P(v_N)\big)$$ 
is closed in $W^k(M,F)$. Take in particular $k=0$. Since $\cC^\infty(M,F)$ is dense
in $W^d(M,F)$, we see that in $W^0(M,F)$ 
$$\left(\wt P\big(W^d(M,F)\big)\right)^\perp=
\Big(P\big(\cC^\infty(M,F)\big)\Big)^\perp=\ker\wt{P^\star}.$$
We have proved that
$$W^0(M,F)=\wt P\big(W^d(M,F)\big)\oplus\ker\wt{P^\star}.
\leqno(2.6)$$
Since $P^\star$ is also elliptic, it follows that $\ker\wt{P^\star}$
is finite dimensional and that $\ker\wt{P^\star}=\ker P^\star$ is
contained in $\cC^\infty(M,F)$. Thanks to G\aa rding's inequality, the
decomposition formula (2.6) yields
$$\leqalignno{
W^k(M,F)&=\wt P\big(W^{k+d}(M,F)\big)\oplus\ker P^\star,&(2.7)\cr
\cC^\infty(M,F)&=P\big(\cC^\infty(M,F)\big)\oplus\ker P^\star.&(2.8)\cr}$$
\endproof

\section{3}{Harmonic Forms and Hodge Theory on Riemannian Manifolds}
\subsection{3.1}{Euclidean Structure of the Exterior Algebra}
Let $(M,g)$ be an oriented Riemannian $\cC^\infty$-manifold, $\dim_{\bR} M=m$,
and $E\to M$ a hermitian vector bundle of rank $r$ over $M$. We denote 
respectively by $(\xi_1,\ldots,\xi_m)$ and $(e_1,\ldots,e_r)$ orthonormal frames of
$T_M$ and $E$ over an open subset $\Omega\subset M$, and by 
$(\xi^\star_1,\ldots,\xi^\star_m)$,~$(e^\star_1,\ldots,e^\star_r)$ the corresponding 
dual frames of $T^\star_M,~E^\star$. Let $dV$ stand for the Riemannian volume
form on $M$. The exterior algebra $\Lambda T^\star_M$ has a natural inner 
product $\langle\bu,\bu\rangle$ such that
$$\langle u_1\wedge\ldots\wedge u_p,v_1\wedge\ldots\wedge v_p\rangle=
\det(\langle u_j,v_k\rangle)_{1\le j,k\le p},~~~~u_j,v_k\in T^\star_M
\leqno(3.1)$$
for all $p$, with $\Lambda T^\star_M=\bigoplus\Lambda^p T^\star_M$ as an
orthogonal sum. Then the covectors $\xi^\star_I=\xi^\star_{i_1}\wedge
\cdots\wedge\xi^\star_{i_p},~
i_1<i_2<\cdots<i_p$, provide an orthonormal basis of $\Lambda T^\star_M$.
We also denote by $\langle\bu,\bu\rangle$ the corresponding inner product on
$\Lambda T^\star_M\otimes E$.

\claim{(3.2) Hodge Star Operator} The Hodge-Poincar\'e-De Rham 
operator $\star$ is the collection of linear maps defined by
$$\star{}:\Lambda^pT^\star_M\to\Lambda^{m-p}T^\star_M,\qquad
u\wedge{}\star v=\langle u,v\rangle\,dV,\qquad
\forall u,v\in\Lambda^pT^\star_M.$$
\endclaim

The existence and uniqueness of this operator is easily seen by using
the duality pairing
$$\leqalignno{
\Lambda^pT^\star_M\times\Lambda^{m-p}T^\star_M&{}\longrightarrow\bR\cr
(u,v)&{}\longmapsto u\wedge v/dV=\sum\varepsilon(I,\complement I)\,
u_Iv_{\complement I},&(3.3)\cr}$$
where $u=\sum_{|I|=p}u_I\,\xi^\star_I$, $v=\sum_{|J|=m-p}v_J\,\xi^\star_J$,
where $\complement I$ stands for the (ordered) complementary multi-index of
$I$ and $\varepsilon(I,\complement I)$ for the signature of the permutation
$(1,2,\ldots,m)\mapsto (I,\complement I)$. From this, we find
$${}\star v=\sum_{|I|=p}\varepsilon(I,\complement I)v_I\,
\xi^\star_{\complement I}.\leqno(3.4)$$
More generally, the sesquilinear pairing $\{\bu,\bu\}$ defined in
(V-7.1) yields an operator $\star$ on vector valued forms, such that
$$\leqalignno{
&\star{}:\Lambda^pT^\star_M\otimes E\to\Lambda^{m-p}T^\star_M\otimes E,\qquad   
\{s,{}\star t\}=\langle s,t\rangle\,dV,\qquad
s,t\in\Lambda^pT^\star_M\otimes E,&(3.3')\cr
&\star t=\sum_{|I|=p,\lambda}\varepsilon(I,\complement I)\,
t_{I,\lambda}\,\xi^\star_{\complement I}\otimes e_\lambda&(3.4')\cr}$$
for $t=\sum t_{I,\lambda}\,\xi^\star_I\otimes e_\lambda$. 
Since $\varepsilon(I,\complement I)\varepsilon(\complement I,I)=(-1)^{p(m-p)}=(-1)^{p(m-1)}$, 
we get immediately
$$\star\star t=(-1)^{p(m-1)}t~~~~{\rm on}~~\Lambda^pT^\star_M\otimes E. 
\leqno(3.5)$$
It is clear that $\star$ is an isometry of $\Lambda^\bu T^\star_M\otimes E$. 

We shall also need a variant of the $\star$ operator, namely the 
conjugate-linear operator
$$\#~:~~\Lambda^pT^\star_M\otimes E\longrightarrow\Lambda^{m-p}T^\star_M\otimes E^\star$$
defined by $s\wedge\#\,t=\langle s,t\rangle\,dV,$
where the wedge product $\wedge$ is combined with the canonical pairing
\hbox{$E\times E^\star\to\bC$}. We have
$$\#\,t=\sum_{|I|=p,\lambda}\varepsilon(I,\complement I)\,\ol t_{I,\lambda}\,
\xi^\star_{\complement I}\otimes e^\star_\lambda.\leqno(3.6)$$

\claim{(3.7) Contraction by a Vector Field.} Given a tangent vector
$\theta\in T_M$ and a form $u\in\Lambda^pT^\star_M$, the
contraction $\theta\ort u\in\Lambda^{p-1}T^\star_M$ is defined by
$$\theta\ort u\,(\eta_1,\ldots,\eta_{p-1})=u(\theta,\eta_1,\ldots,\eta_{p-1}),~~~~
\eta_j\in T_M.$$
\endclaim

In terms of the basis $(\xi_j)$, $\bu\ort\bu$ is the bilinear operation 
characterized by
$$\xi_l\ort(\xi^\star_{i_1}\wedge\ldots\wedge\xi^\star_{i_p})=
\cases{0&if~~$l\notin\{i_1,\ldots,i_p\}$,\cr
(-1)^{k-1}\xi^\star_{i_1}\wedge\ldots\wh{\xi^\star_{i_k}}\ldots\wedge
\xi^\star_{i_p}&if~~$l=i_k$.\cr}$$
This formula is in fact valid even when $(\xi_j)$ is non orthonormal.
A rather easy computation shows that $\theta\ort\bu$ is a {\it derivation}
of the exterior algebra, i.e.\ that
$$\theta\ort(u\wedge v)=(\theta\ort u)\wedge v+(-1)^{{\rm deg}\,u}u\wedge
(\theta\ort v).$$
Moreover, if $\wt\theta=\langle\bu,\theta\rangle\in T^\star_M$, the operator
$\theta\ort\bu$ is the adjoint map of $\wt\theta\wedge\bu$, that is,
$$\langle\theta\ort u,v\rangle=\langle u,\wt\theta\wedge v\rangle,~~~~
u,v\in\Lambda T^\star_M.\leqno(3.8)$$
Indeed, this property is immediately checked when $\theta=\xi_l$,
$u=\xi^\star_I$, $v=\xi^\star_J$.

\subsection{3.2}{Laplace-Beltrami Operators}
Let us consider the Hilbert space $L^2(M,\Lambda^pT^\star_M)$
of $p$-forms $u$ on $M$ with
measurable coefficients such that
$$\|u\|^2=\int_M |u|^2\,dV<+\infty.$$
We denote by $\Ll~~,~\,\Gg$ the global inner product on $L^2$-forms. One
can define similarly the Hilbert space
$L^2(M,\Lambda^pT^\star_M\otimes E)$.

\claim{(3.9) Theorem} The operator $d^\star=(-1)^{mp+1}\star d\,\star $
is the formal adjoint of the exterior derivative $d$ acting on
$\cC^\infty(M,\Lambda^pT^\star_M\otimes E)$.
\endclaim

\proof{Proof} If $u\in \cC^\infty(M,\Lambda^pT^\star_M),~v\in \cC^\infty(M,
\Lambda^{p+1}T^\star_M\otimes)$ are compactly supported we get
$$\eqalign{
\Ll du,v\Gg&=\int_M\langle du,v\rangle\,dV=\int_M du\wedge{}\star v\cr
&=\int_M  d(u\wedge{}\star v)-(-1)^p u\wedge d\star v=-(-1)^p\int_M
u\wedge d\star v\cr}$$
by Stokes' formula. Therefore (3.4) implies
$$\Ll du,v\Gg=-(-1)^p(-1)^{p(m-1)}\int_M u\wedge\star\star d\star v
=(-1)^{mp+1}\Ll u,\star\,d\star v\Gg.\eqno{\square}$$
\endproof

\claim{(3.10) Remark} \rm If $m$ is even, the formula reduces to 
$d^\star=-\star d\,\star $.
\endclaim

\claim{(3.11) Definition} The operator $\Delta=dd^\star+d^\star d$ is called the
Laplace-Beltrami operator of $M$.
\endclaim

Since $d^\star$ is the adjoint of $d$, the Laplace operator $\Delta$ is
formally self-adjoint, i.e.\ $\Ll\Delta u,v\Gg=\Ll u,\Delta v\Gg$ when
the forms $u,v$ are of class $\cC^\infty$ and compactly supported.

\claim{(3.12) Example} \rm Let $M$ be an open subset of $\bR^m$ and 
$g=\sum^m_{i=1} dx_i^2$. In that case we get
$$\eqalign{
u&=\sum_{|I|=p} u_I dx_I,~~~~
du=\sum_{|I|=p,j} {\partial u_I\over\partial x_j} dx_j\wedge dx_I,\cr
\Ll u,v\Gg&=\int_M\langle u,v\rangle\,dV=\int_M\sum_I u_Iv_I\,dV\cr}$$
One can write $dv=\sum dx_j\wedge(\partial v/\partial x_j)$ where
$\partial v/\partial x_j$ denotes the form $v$ in which all coefficients
$v_I$ are differentiated as $\partial v_I/\partial x_j$.
An integration by parts combined with contraction gives
$$\eqalign{
\Ll d^\star u,v\Gg&=\Ll u,dv\Gg=\int_M\langle u,\sum_j dx_j\wedge 
{\partial v\over\partial x_j}\rangle\,dV\cr
&=\int_M\sum_j\langle {\partial\over\partial x_j}\ort u,~{\partial v\over
\partial x_j}\rangle\,dV=-\int_M\langle\sum_j {\partial\over\partial x_j} 
\ort {\partial u\over\partial x_j},v\rangle\,dV ,\cr
d^\star u&=-\sum_j {\partial\over\partial x_j}\ort
{\partial u\over\partial x_j}=-\sum_{I,j}{\partial u_I\over\partial x_j} 
{\partial\over\partial x_j}\ort dx_I.\cr}$$
We get therefore
$$\eqalign{
dd^\star u&=-\sum_{I,j,k} {\partial^2u_I\over\partial x_j\partial x_k}
dx_k\wedge\Big( {\partial\over\partial x_j}\ort dx_I\Big),\cr 
d^\star du&=-\sum_{I,j,k} {\partial^2u_I\over\partial x_j\partial x_k}
{\partial\over\partial x_j}\ort (dx_k\wedge dx_I).\cr}$$
Since 
$${\partial\over\partial x_j}\ort (dx_k\wedge dx_I)=\Big({\partial\over
\partial x_j}\ort dx_k\Big) dx_I-dx_k\wedge\Big({\partial\over\partial
x_j}\ort dx_I\Big),$$
we obtain
$$\Delta u=-\sum_I\Big(\sum_j {\partial^2u_I\over\partial x^2_j}\Big)
dx_I.$$
In the case of an arbitrary riemannian manifold $(M,g)$ we have
$$\eqalign{
u&=\sum u_I\,\xi^\star_I,\cr
du&=\sum_{I,j}(\xi_j\cdot u_I)\,\xi^\star_j\wedge\xi^\star_I
    +\sum_I u_I\,d\xi^\star_I,\cr
d^\star u&=-\sum_{I,j}(\xi_j\cdot u_I)\,\xi_j\ort\xi^\star_I
    +\sum_{I,K}\alpha_{I,K} u_I\,\xi^\star_K\,,\cr}$$
for some $\cC^\infty$ coefficients $\alpha_{I,K}$,~$|I|=p$,~$|K|=p-1$. It follows
that the principal part of  $\Delta$ is the same as that of the second order
operator
$$u\longmapsto-\sum_I\big(\sum_j\xi^2_j\cdot u_I\big)\xi^\star_I.$$
As a consequence, $\Delta$ is {\it elliptic}.
\endclaim

Assume now that $D_E$ is a hermitian connection on $E$. The formal adjoint
operator of  $D_E$ acting on $\cC^\infty(M,\Lambda^pT^\star_M\otimes E)$ is
$$D_E^\star=(-1)^{mp+1}\star  D_E\star  .\leqno (3.13)$$
Indeed, if $s\in \cC^\infty(M,\Lambda^pT^\star_M\otimes E)$,~
$t\in \cC^\infty(M,\Lambda^{p+1}T^\star_M\otimes E)$ have compact 
support, we get
$$\eqalign{
\Ll D_Es,t\Gg&=\int_M\langle D_Es,t\rangle\,dV=\int_M\{D_Es,{}\star t\}\cr
&=\int_M d\{s,{}\star t\}-(-1)^p\{s,D_E\star t\}
=(-1)^{mp+1}\Ll s,{}\star D_E\star t\Gg.\cr}$$

\claim{(3.14) Definition} The Laplace-Beltrami operator associated to
$D_E$ is the second order operator $\Delta_E=D_ED_E^\star+D_E^\star D_E$.
\endclaim

$\Delta_E$ is a self-adjoint elliptic operator with principal part
$$s\longmapsto-\sum_{I,\lambda}\Big(\sum_j\xi^2_j\cdot s_{I,\lambda}
\Big)\xi^\star_I\otimes e_\lambda.$$

\subsection{3.3}{Harmonic Forms and Hodge Isomorphism}
Let $E$ be a hermitian vector bundle over a {\it compact} Riemannian manifold
$(M,g)$. We assume that $E$ possesses a {\it flat} hermitian connection $D_E$
(this means that $\Theta(D_E)=D^2_E=0$, or equivalently, that $E$ is given
by a representation $\pi_1(M)\to U(r)$, cf.\ \S$\,$V-6). A fundamental
example is of course the trivial bundle $E=M\times\bC$ with the connection
$D_E=d$. Thanks to our flatness assumption, $D_E$ defines a generalized
De Rham complex
$$D_E:\cC^\infty(M,\Lambda^pT^\star_M\otimes E)\longrightarrow
\cC^\infty(M,\Lambda^{p+1}T^\star_M\otimes E).$$
The cohomology groups of this complex will be denoted by $H^p_{DR}(M,E)$.

The space of {\it harmonic forms of degree $p$} with respect to
the Laplace-Beltrami operator $\Delta_E=D_ED_E^\star+D_E^\star D_E$ is
defined by
$$\cH^p(M,E)=\big\{ s\in \cC^\infty(M,\Lambda^pT^\star_M\otimes E)~;~
\Delta_E s=0\big\}.\leqno(3.15)$$
Since $\Ll\Delta_Es,s\Gg=||D_Es||^2+||D_E^\star s||^2$, we see that 
$s\in\cH^p(M,E)$ if and only if $D_Es=D_E^\star s=0$.

\claim{(3.16) Theorem} For any $p$, there exists an orthogonal decomposition
$$\eqalign{
&\cC^\infty(M,\Lambda^pT^\star_M\otimes E)=\cH^p(M,E)\oplus\Im D_E
\oplus\Im D_E^\star,\cr
&\Im D_E=D_E\big(\cC^\infty(M,\Lambda^{p-1}T^\star_M\otimes E)\big),\cr
&\Im D_E^\star=D_E^\star\big(\cC^\infty(M,\Lambda^{p+1}T^\star_M\otimes E)\big).
\cr}$$
\endclaim

\proof{Proof} It is immediate that $\cH^p(M,E)$ is orthogonal to both
subspaces $\Im D_E$ and $\Im D_E^\star$. The orthogonality of these two
subspaces is also clear, thanks to the assumption $D^2_E=0\,$:
$$\Ll D_Es,D_E^\star t\Gg=\Ll D^2_Es,t\Gg=0.$$
We apply now Cor.~2.4 to the elliptic operator $\Delta_E=\Delta^\star_E$
acting on \hbox{$p$-forms,} i.e.\ on the bundle $F=\Lambda^pT^\star_M
\otimes E$. We get
$$\eqalignno{
&\cC^\infty(M,\Lambda^pT^\star_M\otimes E)=\cH^p(M,E)\oplus
\Delta_E\big(\cC^\infty(M,\Lambda^pT^\star_M\otimes E)\big),\cr
&\Im\Delta_E=\Im(D_ED_E^\star+D_E^\star D_E)
\subset \Im D_E+\Im D_E^\star.&\square\cr}$$
\endproof

\claim{(3.17) Hodge isomorphism theorem} The De Rham cohomology group
$H^p_{DR}(M,E)$ is finite dimensional and $H^p_{DR}(M,E)\simeq\cH^p(M,E)$.
\endclaim

\proof{Proof} According to decomposition 3.16, we get
$$\eqalign{
B^p_{DR}(M,E)&=D_E\big(\cC^\infty(M,\Lambda^{p-1}T^\star_M\otimes E)\big),\cr
Z^p_{DR}(M,E)&=\ker D_E=(\Im D_E^\star)^\perp=\cH^p(M,E)\oplus\Im D_E.\cr}$$
This shows that every De Rham cohomology class contains a unique harmonic
representative.\qed
\endproof

\claim{(3.18) Poincar\'e duality} The bilinear pairing
$$H^p_{DR}(M,E)\times H^{m-p}_{DR}(M,E^\star)\longrightarrow\bC,~~~~
(s,t)\longmapsto\int_M s\wedge t$$
is a non degenerate duality.
\endclaim

\proof{Proof} First note that there exists a naturally defined flat connection
$D_{E^\star}$ such that for any $s_1\in \cC^\infty_\bu(M,E)$,
$s_2\in \cC^\infty_\bu(M,E^\star)$ we have
$$d(s_1\wedge s_2)=(D_Es_1)\wedge s_2+(-1)^{\deg s_1}s_1\wedge D_{E^\star}s_2.
\leqno(3.19)$$
It is then a consequence of Stokes' formula that the map $(s,t)\mapsto\int_M
s\wedge t$ can be factorized through cohomology groups. Let $s\in 
\cC^\infty(M,\Lambda^pT^\star_M\otimes E)$. We leave to the reader the proof
of the following formulas (use (3.19) in analogy with the proof of Th.~3.9):
$$\leqalignno{
D_{E^\star}(\#\,s)&=(-1)^p\#\,D_E^\star s,\cr
\delta_{E^\star}(\#\,s)&=(-1)^{p+1}\#\,D_E s,&(3.20)\cr
\Delta_{E^\star}(\#\,s)&=\#\,\Delta_E s,\cr}$$
Consequently $\#s\in\cH^{m-p}(M,E^\star)$ if and only if $s\in\cH^p(M,E)$. Since
$$\int_M s\wedge\#\,s=\int_M |s|^2\,dV=\|s\|^2,$$
we see that the Poincar\'e pairing has zero kernel in the left hand factor
$\cH^p(M,E)\simeq H^p_{DR}(M,E)$. By symmetry, it has also zero kernel on
the right. The proof is achieved.\qed
\endproof

\section{4}{Hermitian and K\"ahler Manifolds}
Let $X$ be a complex $n$-dimensional manifold. A {\it hermitian metric}
on $X$ is a positive definite hermitian form of class $\cC^\infty$ on $T_X\,$;
in a coordinate system $(z_1,\ldots,z_n)$, such a form can be written
\hbox{$h(z)=\sum_{1\le j,k\le n} h_{jk}(z)\,dz_j\otimes d\ol z_k$,}
where $(h_{jk})$ is a positive hermitian matrix with $\cC^\infty$ coefficients.
According to~(III-1.8), the {\it fundamental $(1,1)$-form} associated to $h$ is
the positive form of type $(1,1)$
$$\omega=-{\rm Im~} h={\ii\over 2}\sum~ h_{jk} dz_j\wedge d\ol z_k,~~~~
1\le j,k\le n.$$

\claim{(4.1) Definition} \smallskip
\item{\rm a)} A hermitian manifold is a pair  $(X,\omega)$ where $\omega$
is a $\cC^\infty$ positive definite $(1,1)$-form on $X$.
\smallskip
\item{\rm b)} The metric $\omega$ is said to be k\"ahler if $d\omega=0$.
\smallskip
\item{\rm c)} $X$ is said to be a K\"ahler manifold if $X$ carries at
least one K\"ahler metric.\smallskip
\endclaim

Since $\omega$ is real, the conditions $d\omega=0$, $d'\omega=0$,
$d''\omega=0$ are all equivalent. In local coordinates we see that
$d'\omega=0$ if and only if
$${\partial h_{jk}\over\partial z_l}={\partial h_{lk}\over
\partial z_j}\quad,\quad 1\le j,k,l\le n.$$
A simple computation gives
$${\omega^n\over n!}
=\det(h_{jk})\bigwedge_{1\le j\le n}\Big({\ii\over 2} dz_j\wedge d\ol z_j\Big)
=\det(h_{jk})\,dx_1\wedge dy_1\wedge\cdots\wedge dx_n\wedge dy_n,$$
where $z_n=x_n+\ii y_n$. Therefore the $(n,n)$-form
$$dV={1\over n!}\omega^n\leqno(4.2)$$
is positive and coincides with the hermitian volume element of $X$. If $X$ is
compact, then $\int_X\omega^n=n!\,\Vol_\omega(X)>0$. This simple remark
already implies that compact K\"ahler manifolds must satisfy some restrictive
topological conditions:

\claim{(4.3) Consequence} \smallskip
\item{\rm a)} If $(X,\omega)$ is compact K\"ahler and if $\{\omega\}$
denotes the  cohomology class of $\omega$ in {$H^2(X,\bR),\kern-1.8pt$} then
$\{\omega\}^n\ne 0$.
\smallskip
\item{\rm b)} If $X$ is compact K\"ahler, then $H^{2k}(X,\bR)\ne 0$ for
$0\le k\le n$. In fact, $\{\omega\}^k$ is a non zero class in
$H^{2k}(X,\bR)$.
\endclaim

\claim{(4.4) Example} \rm The complex projective space $\bP^n$ is K\"ahler.
A natural K\"ahler metric $\omega$ on $\bP^n$, called the {\it Fubini-Study
metric}, is defined by
$$p^\star\omega={\ii\over 2\pi} d'd''\log\big(|\zeta_0|^2+|\zeta_1|^2
+\cdots+|\zeta_n|^2\big)$$
where $\zeta_0,\zeta_1,\ldots,\zeta_n$ are coordinates of $\bC^{n+1}$ and
where \hbox{$p:\bC^{n+1}\setminus\{0\}\to\bP^n$} is the projection.
Let $z=(\zeta_1/\zeta_0,\ldots,\zeta_n/\zeta_0)$ be non homogeneous coordinates
on $\bC^n\subset\bP^n$. Then (V-15.8) and (V-15.12) show that
$$\omega={\ii\over 2\pi}d'd''\log(1+|z|^2)={\ii\over 2\pi}c\big(\cO(1)\big),
~~~~\int_{\bP^n}\omega^n=1.$$
Furthermore $\{\omega\}\in H^2(\bP^n,\bZ)$ is a generator of the
cohomology algebra $H^\bu(\bP^n,\bZ)$ in virtue of Th.~V-15.10.
\endclaim

\claim{(4.5) Example} \rm A {\it complex torus} is a quotient
$X=\bC^n/\Gamma$ by a lattice $\Gamma$ of rank~$2n$. Then $X$ is a
compact complex manifold. Any positive definite hermitian form
$\omega=\ii\sum h_{jk}dz_j\wedge d\ol z_k$ with constant coefficients
defines a K\"ahler metric on~$X$.
\endclaim

\claim{(4.6) Example} \rm Every (complex) submanifold $Y$ of a
K\"ahler manifold  $(X,\omega)$ is K\"ahler with metric
$\omega_{\restriction Y}$.  Especially, all submanifolds of $\bP^n$
are K\"ahler.
\endclaim

\claim{(4.7) Example} \rm Consider the complex surface
$$X=(\bC^2\setminus\{ 0\})/\Gamma$$
where $\Gamma=\{\lambda^n~;~n\in\bZ\}$, $\lambda<1$, acts as a
group of homotheties. Since $\bC^2\setminus\{ 0\}$ is diffeomorphic
to  $\bR^\star_+\times S^3$, we have $X\simeq S^1\times S^3$. 
Therefore $H^2(X,\bR)=0$ by K\"unneth's formula IV-15.10, and
property 4.3 b) shows that $X$ is not K\"ahler. More generally,
one can obtaintake $\Gamma$ to be an infinite cyclic group generated by
a holomorphic contraction of $\bC^2$, of the form
$$\pmatrix{z_1\cr z_2\cr}\longmapsto\pmatrix{\lambda_1z_1\cr\lambda_2z_2\cr},
\qquad\hbox{\rm resp.}\quad\pmatrix{z_1\cr z_2\cr}\longmapsto
\pmatrix{\lambda z_1\cr\lambda z_2+z_1^p\cr},$$
where $\lambda,\lambda_1,\lambda_2$ are complex numbers such that
$0<|\lambda_1|\le|\lambda_2|<1$, \hbox{$0<|\lambda|<1$}, and $p$ a positive
integer. These non K\"ahler surfaces are called {\it Hopf surfaces}.
\endclaim

The following Theorem shows that a hermitian metric $\omega$ on $X$
is K\"ahler if and only if the metric $\omega$ is tangent at order $2$ to a 
hermitian metric with constant coefficients at every point of~$X$.

\claim{(4.8) Theorem} Let $\omega$ be a $\cC^\infty$ positive definite $(1,1)$-form
on $X$. In order that $\omega$ be K\"ahler, it is necessary and sufficient
that to every point $x_0\in X$ corresponds a holomorphic coordinate system
$(z_1,\ldots,z_n)$ centered at $x_0$ such that
$$\omega=\ii\sum_{1\le l,m\le n}\omega_{lm}\,dz_l\wedge d\ol z_m,~~~~
\omega_{lm}=\delta_{lm}+O(|z|^2).\leqno(4.9)$$
If $\omega$ is K\"ahler, the coordinates $(z_j)_{1\le j\le n}$ can be
chosen such that
$$\omega_{lm}=\langle{\partial\over\partial z_l},{\partial\over\partial z_m}
\rangle=\delta_{lm}-\sum_{1\le j,k\le n}c_{jklm}\,z_j\ol z_k+O(|z|^3)
,\leqno(4.10)$$
where $(c_{jklm})$ are the coefficients of the Chern curvature tensor
$$\Theta(T_X)_{x_0}=\sum_{j,k,l,m} c_{jklm}\,dz_j\wedge d\ol z_k\otimes
\Big({\partial\over\partial z_l}\Big)^\star
\otimes{\partial\over\partial z_m}\leqno(4.11)$$
associated to $(T_X,\omega)$ at $x_0$. Such a system $(z_j)$
will be called a geodesic coordinate system at $x_0$.
\endclaim

\proof{Proof} It is clear that (4.9) implies $d_{x_0}\omega=0$, so the
condition is sufficient. Assume now that $\omega$ is K\"ahler. Then one can
choose local coordinates $(x_1,\ldots,x_n)$ such that $(dx_1,\ldots,dx_n)$ is an
$\omega$-orthonormal basis of $T_{x_0}^\star X$. Therefore
$$\leqalignno{
\omega&=\ii\sum_{1\le l,m\le n}\wt\omega_{lm}\,dx_l\wedge d\ol x_m,
~~~~{\rm where}\cr
\qquad\wt\omega_{lm}&=\delta_{lm}+O(|x|)=\delta_{lm}+\sum_{1\le j\le n}
(a_{jlm}x_j+a'_{jlm}\ol x_j)+O(|x|^2).&(4.12)\cr}$$
Since $\omega$ is real, we have $a'_{jlm}=\ol a_{jml}\,$; on the other
hand the K\"ahler condition $\partial\omega_{lm}/\partial x_j=
\partial\omega_{jm}/\partial x_l$ at $x_0$ implies $a_{jlm}=a_{ljm}$. Set now
$$z_m=x_m+{1\over 2}\sum_{j,l}a_{jlm}x_jx_l,~~~~1\le m\le n.$$
Then $(z_m)$ is a coordinate system at $x_0$, and
$$\eqalign{
dz_m&=dx_m+\sum_{j,l}a_{jlm}x_jdx_l,\cr
\ii\sum_mdz_m\wedge d\ol z_m&=\ii\sum_m dx_m\wedge d\ol x_m
+\ii\sum_{j,l,m}a_{jlm}x_j\,dx_l\wedge d\ol x_m\cr
&\phantom{=\ii\sum_m dx_m~\wedge d\ol x_m}
+\ii\sum_{j,l,m}\ol a_{jlm}\ol x_j\,dx_m\wedge d\ol x_l+O(|x|^2)\cr
&=\ii\sum_{l,m}\wt\omega_{lm}\,dx_l\wedge\ol dx_m+O(|x|^2)=
\omega+O(|z|^2).\cr}$$
Condition (4.9) is proved. Suppose the coordinates $(x_m)$
chosen from the beginning so that (4.9) holds with respect to $(x_m)$.
Then the Taylor expansion (4.12) can be refined into
$$\leqalignno{
\qquad\wt\omega_{lm}&=\delta_{lm}+O(|x|^2)&(4.13)\cr
&=\delta_{lm}+\sum_{j,k}
\big(a_{jklm}x_j\ol x_k+a'_{jklm}x_jx_k+a''_{jklm}\ol x_j\ol x_k\big)+
O(|x|^3).}$$
These new coefficients satisfy the relations
$$a'_{jklm}=a'_{kjlm},~~~~a''_{jklm}=\ol a'_{jkml},~~~~\ol a_{jklm}=a_{kjml}.$$
The K\"ahler condition $\partial\omega_{lm}/\partial x_j=\partial
\omega_{jm}/\partial x_l$ at $x=0$ gives the equality
$a'_{jklm}=a'_{lkjm}\,;$ in particular $a'_{jklm}$ is invariant under all
permutations of~$j,k,l$. If we set
$$z_m=x_m+{1\over 3}\sum_{j,k,l}a'_{jklm}\,x_jx_kx_l,~~~~1\le m\le n,$$
then by (4.13) we find
$$\leqalignno{
\qquad dz_m&=dx_m+\sum_{j,k,l}a'_{jklm}\,x_jx_k\,dx_l,~~~~1\le m\le n,\cr
\omega&=\ii\sum_{1\le m\le n}dz_m\wedge d\ol z_m+\ii\sum_{j,k,l,m}a_{jklm}\,x_j
\ol x_k\,dx_l\wedge d\ol x_m+O(|x|^3),\cr
\omega&=\ii\sum_{1\le m\le n}dz_m\wedge d\ol z_m+\ii\sum_{j,k,l,m}a_{jklm}\,z_j
\ol z_k\,dz_l\wedge d\ol z_m+O(|z|^3).&(4.14)\cr}$$
It is now easy to compute the Chern curvature tensor $\Theta(T_X)_{x_0}$ in terms of
the coefficients $a_{jklm}$. Indeed
$$\eqalign{
\langle{\partial\over\partial z_l},{\partial\over\partial z_m}\rangle&=
\delta_{lm}+\sum_{j,k}a_{jklm}\,z_j\ol z_k+O(|z|^3),\cr
d'\langle{\partial\over\partial z_l},{\partial\over\partial z_m}\rangle&=
\Big\{D'{\partial\over\partial z_l},{\partial\over\partial z_m}\Big\}=
\sum_{j,k}a_{jklm}\,\ol z_k\,dz_j+O(|z|^2),\cr
\Theta(T_X)\cdot{\partial\over\partial z_l}&=D''D'\Big({\partial\over\partial z_l}\Big)
=-\sum_{j,k,m}a_{jklm}\,dz_j\wedge d\ol z_k\otimes{\partial\over\partial z_m}
+O(|z|),\cr}$$
therefore $c_{jklm}=-a_{jklm}$ and the expansion (4.10) follows from 
(4.14).\qed
\endproof

\claim{(4.15) Remark} \rm As a by-product of our computations, we
find that on a K\"ahler manifold the coefficients of $\Theta(T_X)$ satisfy
the symmetry relations
$$\ol c_{jklm}=c_{kjml},~~~~c_{jklm}=c_{lkjm}=c_{jmlk}=c_{lmjk}.$$
\endclaim

\section{5}{Basic Results of K\"ahler Geometry}
\subsection{5.1}{Operators of Hermitian Geometry}
Let $(X,\omega)$ be a hermitian manifold and let $z_j=x_j+\ii y_j$,
$1\le j\le n$, be analytic coordinates at a point $x\in X$ such that
$\omega(x)=\ii\sum dz_j\wedge d\ol z_j$ is diagonalized at this point.
The associated hermitian form is the $h(x)=2\sum dz_j\otimes d\ol z_j$
and its real part is the euclidean metric $2\sum (dx_j)^2+(dy_j)^2$. It
follows from this that $|dx_j|=|dy_j|=1/\sqrt{2}$, $|dz_j|=|d\ol z_j|=1$,
and that $(\partial/\partial z_1,\ldots,\partial/\partial z_n)$ is an orthonormal
basis of $(T^\star_x X,\omega)$. Formula (3.1) with $u_j,v_k$ in the
orthogonal sum $(\bC\otimes T_X)^\star=T^\star_X\oplus\ol{T^\star_X}$
defines a natural inner product on the exterior algebra
$\Lambda^\bu(\bC\otimes T_X)^\star$. The norm of a form
$$u=\sum_{I,J}~~u_{I,J} dz_I\wedge d\ol z_J\in\Lambda
(\bC\otimes T_X)^\star$$
at the given point $x$ is then equal to
$$|u(x)|^2=\sum_{I,J} ~|u_{I,J}(x)|^2.\leqno(5.1)$$

The Hodge $\star $ operator (3.2) can be extended to $\bC$-valued forms by 
the formula
$$u\wedge{}\star\ol v=\langle u,v\rangle\,dV.\leqno(5.2)$$
It follows that $\star $ is a $\bC$-linear isometry
$$\star~:~~\Lambda^{p,q} T^\star_X\longrightarrow\Lambda^{n-q,n-p} T^\star_X.$$
The usual operators of hermitian geometry are the operators $d,~\delta=
-\star d\,\star ,~\Delta=d\delta+\delta d$ already defined, and their
complex counterparts
$$\left\{\eqalign{
d&=d'+d'',\cr
\delta&=d^{\prime\star}+d^{\prime\prime\star},~~~~d^{\prime\star}=(d')^\star=-\star d''\star,~~~~
d^{\prime\prime\star}=(d'')^\star=-\star d'\star,\cr
\Delta'&=d'd^{\prime\star}+d^{\prime\star}d',~~~~
\Delta''=d''d^{\prime\prime\star}+d^{\prime\prime\star}d''.\cr}\right.\leqno(5.3)$$
Another important operator is the operator $L$ of type (1,1) defined by
$$Lu=\omega\wedge u\leqno(5.4)$$
and its adjoint  $\Lambda=\star ^{-1}L\,\star ~:$
$$\langle u,\Lambda v\rangle=\langle Lu,v\rangle.\leqno(5.5)$$

\subsection{5.2}{Commutation Identities}
If $A,B$ are endomorphisms of the algebra $\cC^\infty_{\bu,\bu}(X,\bC)$,
their graded commutator (or graded Lie bracket) is defined by
$$[A,B]=AB-(-1)^{ab} BA\leqno(5.6)$$
where $a,b$ are the degrees of $A$ and $B$ respectively. If $C$ is another
endomorphism of degree $c$, the following {\it Jacobi identity} is easy to
check:
$$(-1)^{ca}\big[A,[B,C]\big]+(-1)^{ab}\big[B,[C,A]\big]
+(-1)^{bc}\big[C,[A,B]\big]=0.\leqno(5.7)$$
For any $\alpha\in\Lambda^{p,q}T^\star_X$, we still denote by $\alpha$
the endomorphism of type $(p,q)$ on $\Lambda^{\bu,\bu} T^\star_X$ defined by
$u\mapsto\alpha\wedge u$. Consider further  a real (1,1)-form
$$\gamma\in\Lambda^{1,1} T^\star_X.$$
There exists an
$\omega$-orthogonal basis $(\zeta_1,\zeta_2,\ldots,\zeta_n)$ in $T_X$ which
diagonalizes both forms $\omega$ and $\gamma\,$:
$$\omega=\ii\sum_{1\le j\le n}\zeta^\star_j\wedge\ol\zeta^\star_j,~~~~
\gamma=\ii\sum_{1\le j\le n}\gamma_j\,\zeta^\star_j\wedge\ol\zeta^\star_j,~~
\gamma_j\in\bR.$$

\claim{(5.8) Proposition} For every form $u=\sum u_{J,K}\,\zeta^\star_J 
\wedge\ol\zeta^\star_K$, one has
$$[\gamma,\Lambda]u=\sum_{J,K}\Big(\sum_{j\in J}\gamma_j+\sum_{j\in K}
\gamma_j-\sum_{1\le j\le n}\gamma_j\Big) u_{J,K}\,\zeta^\star_J\wedge\ol\zeta^\star_K.$$
\endclaim

\proof{Proof} If $u$ is of type $(p,q)$, a brute-force computation yields
$$\eqalignno{
\Lambda u&=\ii(-1)^p\sum_{J,K,l} u_{J,K}\,(\zeta_l\ort\zeta^\star_J)
\wedge(\ol\zeta_l\ort\ol\zeta^\star_K),~~~~1\le l\le n,\cr 
\gamma\wedge u&=\ii(-1)^p\sum_{J,K,m}\gamma_m u_{J,K}\,\zeta^\star_m\wedge
\zeta^\star_J\wedge\ol\zeta^\star_m\wedge\ol\zeta^\star_K,~~~~1\le m\le n,\cr 
[\gamma,\Lambda] u&=\sum_{J,K,l,m}\gamma_m\,u_{J,K}\Big(
\big(\zeta^\star_l\wedge(\zeta_m\ort\zeta^\star_J)\big)\wedge
\big(\ol\zeta^\star_l\wedge(\ol\zeta_m\ort\ol\zeta^\star_K)\big)\cr
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-\big(\zeta_m\ort(\zeta^\star_l\wedge\zeta^\star_J)\big)\wedge
\big(\ol\zeta_m\ort(\ol\zeta^\star_l\wedge\ol\zeta^\star_K)\big)\Big)\cr
&=\sum_{J,K,m}\gamma_m\,u_{J,K}\Big(
\zeta^\star_m\wedge(\zeta_m\ort\zeta^\star_J)\wedge\ol\zeta^\star_K\cr
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+\zeta^\star_J\wedge\ol\zeta^\star_m\wedge(\ol\zeta_m\ort\ol\zeta^\star_K)-
\zeta^\star_J\wedge\ol\zeta^\star_K\Big)\cr
&=\sum_{J,K}\bigg(\sum_{m\in J}\gamma_m+\sum_{m\in K}\gamma_m-\sum_{1\le 
m\le n}\gamma_m\bigg) u_{J,K}\,\zeta^\star_J\wedge\ol\zeta^\star_K.
&\square\cr}$$
\endproof

\claim{(5.9) Corollary} For every $u\in\Lambda^{p,q}T^\star_X$, we have
$$[L,\Lambda]u=(p+q-n) u.$$
\endclaim

\proof{Proof} Indeed, if $\gamma=\omega$, we have $\gamma_1=\cdots=\gamma_n=1$.
\qed
\endproof

This result can be generalized as follows: for every
$u\in\Lambda^k(\bC\otimes T_X)^\star$, we have
$$[L^r,\Lambda]u=r(k-n+r-1)\,L^{r-1}u.\leqno(5.10)$$
In fact, it is clear that
$$\eqalign{
[L^r,\Lambda]u&=\sum_{0\le m\le r-1}\kern-8pt L^{r-1-m}[L,\Lambda]L^m u\cr
&=\sum_{0\le m\le r-1}\kern-8pt (2m+k-n)L^{r-1-m}L^m u
=\big(r(r-1)+r(k-n)\big) L^{r-1} u.\cr}$$

\subsection{5.3}{Primitive Elements and Hard Lefschetz Theorem}
In this subsection, we prove a fundamental decomposition theorem for the
representation of the unitary group $U(T_X)\simeq U(n)$ acting on
the spaces $\Lambda^{p,q}T^\star_X$ of $(p,q)$-forms. It turns out that
the representation is never irreducible if $0<p,q<n$.

\claim{(5.11) Definition} A homogeneous element $u\in\Lambda^k
(\bC\otimes T_X)^\star$
is called {\it primitive} if $\Lambda u=0$. The space of primitive elements
of total degree $k$ will be denoted
$$\Prim^kT^\star_X=\bigoplus_{p+q=k}\Prim^{p,q}T^\star_X.$$
\endclaim

Let  $u\in\Prim^kT^\star_X$. Then
$$\Lambda^s L^r u=\Lambda^{s-1}(\Lambda L^r-L^r\Lambda)u
=r(n-k-r+1)\Lambda^{s-1}L^{r-1}u.$$
By induction, we get for $r\ge s$
$$\Lambda^sL^ru=r(r-1)\cdots (r-s+1)\cdot (n-k-r+1) 
\cdots (n-k-r+s)L^{r-s}u.\leqno(5.12)$$
Apply (5.12) for $r=n+1$. Then $L^{n+1}u$ is of degree $>2n$ and
therefore we have $L^{n+1}u=0$. This gives
$$(n+1)\cdots\big(n+1-(s-1)\big)\cdot (-k)(-k+1)\cdots(-k+s-1)L^{n+1-s}u=0.$$
The integral coefficient is $\ne 0$ if $s\le k$, hence:

\claim{(5.13) Corollary} If $u\in\Prim^kT^\star_X$, then $L^su=0$ for
$s\ge (n+1-k)_+$.
\endclaim

\claim{(5.14) Corollary} $\Prim^kT^\star_X=0$ for $n+1\le k\le 2n$.
\endclaim

\proof{Proof} Apply Corollary 5.13 with $s=0$.\qed
\endproof

\claim{(5.15) Primitive decomposition formula} For every
$u\in\Lambda^k(\bC\otimes T_X)^\star$, there is a unique decomposition
$$u=\sum_{r\ge (k-n)_+} L^ru_r,~~~~u_r\in\Prim^{k-2r}T^\star_X.$$
Furthermore~ $u_r=\Phi_{k,r} (L,\Lambda)u$~ where $\,\Phi_{k,r}\,$ is a non
commutative polynomial in $L,\Lambda$ with rational coefficients. As
a consequence, there are direct sum decompositions of $U(n)$-representations
$$\eqalign{
&\Lambda^k(\bC\otimes T_X)^\star=\bigoplus_{r\ge (k-n)_+}
L^r\Prim^{k-2r}T^\star_X,\cr
&\Lambda^{p,q}T_X^\star=\bigoplus_{r\ge(p+q-n)_+}
L^r\Prim^{p-r,q-r}T^\star_X.\cr}$$
\endclaim

\proof{Proof of the uniqueness of the decomposition}
Assume that $u=0$ and that $u_r\ne 0$ 
for some $r$. Let $s$ be the largest integer such that $u_s\ne 0$. Then
$$\Lambda^s u=0=\sum_{(k-n)_+\le r\le s}\Lambda^s L^r u_r=
\sum_{(k-n)_+\le r\le s}\Lambda^{s-r}\Lambda^r L^r u_r.$$
But formula (5.12) shows that $\Lambda^r L^r u_r=c_{k,r} u_r$ for
some non zero integral coefficient $c_{k,r}=r!(n-k+r+1)\cdots (n-k+2r)$.
Since $u_r$ is primitive we get $\Lambda^sL^ru_r=0$ when $r<s$, hence
$u_s=0$, a contradiction.
\endproof

\proof{Proof of the existence of the decomposition} We prove by induction on
$s\ge(k-n)_+$ that $\Lambda^s u=0$ implies
$$u=\sum_{(k-n)_+\le r<s} L^ru_r,~~~~
u_r=\Phi_{k,r,s}(L,\Lambda)u\in\Prim^{k-2r}T^\star_X.\leqno(5.16)$$
The Theorem will follow from the step $s=n+1$.

Assume that the
result is true for $s$ and that $\Lambda^{s+1}u=0$. Then $\Lambda^s u$ is
in $\Prim^{k-2s}T^\star_X$. Since $s\ge (k-n)_+$ we have $c_{k,s}\ne 0$
and we set
$$\eqalign{
u_s&={1\over c_{k,s}}\Lambda^s u\in\Prim^{k-2s}T^\star_X,\cr
u'&=u-L^s u_s=\Big(1-{1\over c_{k,s}}L^s\Lambda^s\Big)u.\cr}$$
By formula (5.12), we get
$$\Lambda^s u'=\Lambda^su-\Lambda^s L^s u_s=\Lambda^s u-c_{k,s} u_s=0.$$
The induction hypothesis implies
$$u'=\sum_{(k-n)_+\le r<s} L^ru'_r,~u'_r=\Phi_{k,r,s} (L,\Lambda)
u'\in\Prim^{k-2r}T^\star_X,$$
hence
$u=\sum_{(k-n)_+\le r\le s} L^ru_r$ with
$$\cases{
u_r=u'_r=\Phi_{k,r,s}(L,\Lambda)\Big(1-{\displaystyle 1\over\displaystyle 
c_{k,s}} L^s\Lambda^s\Big)u,~r<s,\cr
u_s={\displaystyle 1\over\displaystyle c_{k,s}}\Lambda^su.\cr}$$
It remains to prove the validity of the decomposition 5.16) for the initial 
step $s=(k-n)_+$, i.e.\ that $\Lambda^su=0$ implies $u=0$. If $k\le n$, then $s=0$ and there is nothing to prove.
We are left with the
case  $k>n$, $\Lambda^{k-n}u=0$. Then $v={}\star u\in\Lambda^{2n-k}
(\bC\otimes T_X)^\star$ and $2n-k<n$. Since the decomposition exists in degree
$\le n$ by what we have just proved, we get
$$\eqalign{
v&={}\star u=\sum_{r\ge 0} L^rv_r,~ v_r\in\Prim^{2n-k-2r}T^\star_X,\cr
0&={}\star\Lambda^{k-n}u=L^{k-n}\star  u=\sum_{r\ge 0} L^{r+k-n} v_r,\cr}$$
with degree $(L^{r+k-n}v_r)=2n-k+2(k-n)=k$. The uniqueness part shows that
$v_r=0$ for~all~$r\,$, hence $u=0$. The Theorem is proved.\qed
\endproof

\claim{(5.17) Corollary} The linear operators
$$\eqalign{
L^{n-k}:\Lambda^k (\bC\otimes T_X)^\star
&\longrightarrow\Lambda^{2n-k}(\bC\otimes T_X)^\star,\cr
L^{n-p-q}:\Lambda^{p,q}T^\star_X&\longrightarrow\Lambda^{n-q,n-p}T^\star_X,\cr}$$
are isomorphisms for all integers $k\le n$, $p+q\le n$.
\endclaim

\proof{Proof} For every $u\in\Lambda^k_\bC T^\star_X$, the primitive
decomposition $u=\sum_{r\ge 0} L^ru_r$ is mapped bijectively onto
that of $L^{n-k}u\,$:
$$L^{n-k}u=\sum_{r\ge 0} L^{r+n-k}u_r.\eqno{\square}$$
\endproof

\section{6}{Commutation Relations}
\subsection{6.1}{Commutation Relations on a K\"ahler Manifold}
Assume first that $X=\Omega\subset\bC^n$ is an open subset and that
$\omega$ is the standard K\"ahler metric
$$\omega=\ii\sum_{1\le j\le n} dz_j\wedge d\ol z_j.$$
For any form $u\in \cC^\infty(\Omega,\Lambda^{p,q}T^\star_X)$ we have
$$\leqalignno{
d'u&=\sum_{I,J,k} {\partial u_{I,J}\over\partial z_k}
dz_k\wedge dz_I\wedge d\ol z_J ,&(6.1')\cr
d''u&=\sum_{I,J,k} {\partial u_{I,J}\over\partial\ol z_k}
d\ol z_k\wedge dz_I\wedge d\ol z_J.&(6.1'')\cr}$$
Since the global $L^2$ inner product is given by
$$\Ll u,v\Gg=\int_\Omega\sum_{I,J} u_{I,J}\ol v_{I,J}\,dV,$$
easy computations analogous to those of Example 3.12 show that
$$\leqalignno{
d^{\prime\star}u&=-\sum_{I,J,k} {\partial u_{I,J}\over\partial\ol z_k}
{\partial\over\partial z_k}\ort (dz_I\wedge d\ol z_J),&(6.2')\cr
d^{\prime\prime\star}u&=-\sum_{I,J,k} {\partial u_{I,J}\over\partial z_k}
{\partial\over\partial\ol z_k}\ort (dz_I\wedge d\ol z_J).&(6.2'')\cr}$$
We first prove a lemma due to \cite{Akizuki and Nakano 1954}.

\claim{(6.3) Lemma} In $\bC^n$, we have $[d^{\prime\prime\star},L]=\ii d'$.
\endclaim

\proof{Proof} Formula (6.$2''$) can be written more briefly
$$d^{\prime\prime\star}u=-\sum_k {\partial\over\partial\ol z_k}\ort\Big(
{\partial u\over\partial z_k}\Big).$$
Then we get
$$[d^{\prime\prime\star},L]u=-\sum_k {\partial\over\partial\ol z_k}\ort\Big(
{\partial\over\partial z_k}(\omega\wedge u)\Big)+\omega\wedge\sum_k 
{\partial\over\partial\ol z_k}\ort\Big({\partial u\over\partial z_k}\Big).$$
Since $\omega$ has constant coefficients, we have $\displaystyle{\partial\over
\partial z_k}(\omega\wedge u)=\omega\wedge{\partial u\over\partial z_k}$
and therefore
$$\eqalign{
[d^{\prime\prime\star},L]\,u&=-\sum_k\bigg({\partial\over\partial\ol z_k}\ort
\Big(\omega\wedge {\partial u\over\partial z_k}\Big)-\omega\wedge\Big(
{\partial\over\partial\ol z_k}\ort{\partial u\over\partial z_k}\Big)\bigg)\cr
&=-\sum_k\Big({\partial\over\partial\ol z_k}\ort\omega\Big)\wedge
{\partial u\over\partial z_k}.\cr}$$
Clearly $\displaystyle{\partial\over\partial\ol z_k}\ort\omega=-\ii dz_k$, so
$$[d^{\prime\prime\star},L]\,u=\ii\sum_k dz_k\wedge
{\partial u\over\partial z_k}=\ii d'u.
\eqno{\square}$$
\endproof

We are now ready to derive the basic commutation relations in the case of an
arbitrary K\"ahler manifold $(X,\omega)$.

\claim{(6.4) Theorem} If $(X,\omega)$ is K\"ahler, then
$$\cmalign{&[d^{\prime\prime\star},L]&=&\ii d',~~~~~
&[d^{\prime\star},L]&=-&\ii d'',\cr
&[\Lambda,d'']&=-&\ii d^{\prime\star},~~~~~
&[\Lambda,d']&=&\ii d^{\prime\prime\star}.\cr}$$
\endclaim

\proof{Proof} It is sufficient to verify the first relation, because the
second one is the conjugate of the first, and the relations of the second line
are the adjoint of those of the first line. If $(z_j)$ is a geodesic
coordinate system at a point $x_0\in X$, then for any $(p,q)$-forms $u,v$
with compact support in a neighborhood of $x_0$, (4.9) implies
$$\Ll u,v\Gg=\int_M\Big(\sum_{I,J}u_{IJ}\ol v_{IJ}+\sum_{I,J,K,L}a_{IJKL}\,
u_{IJ}\ol v_{KL}\Big)\,dV,$$
with $a_{IJKL}(z)=O(|z|^2)$ at $x_0$. An integration by parts as in 
(3.12) and (6.$2''$) yields
$$d^{\prime\prime\star}u=-\sum_{I,J,k} {\partial u_{I,J}\over\partial z_k}
{\partial\over\partial\ol z_k}\ort(dz_I\wedge d\ol z_J)+
\sum_{I,J,K,L}b_{IJKL}\,u_{IJ}\,dz_K\wedge d\ol z_L,$$
where the coefficients~ $b_{IJKL}$~ are obtained by derivation of the~
$a_{IJKL}$'s. Therefore $b_{IJKL}=O(|z|)$. Since
$\partial\omega/\partial z_k=O(|z|)$, the proof of Lemma~6.3 implies
here  $[d^{\prime\prime\star},L]u=\ii d'u+O(|z|)$, in particular
both terms coincide at every given point $x_0\in X$.\qed
\endproof

\claim{(6.5) Corollary} If $(X,\omega)$ is K\"ahler, the complex
Laplace-Beltrami operators satisfy 
$$\Delta'=\Delta''={1\over 2}\Delta.$$
\endclaim

\proof{Proof} It will be first shown that $\Delta''=\Delta'$. We have
$$\Delta''=[d'',d^{\prime\prime\star}]=-\ii\big[d'',[\Lambda,d']\big].$$
Since $[d',d'']=0$, Jacobi's identity (5.7) implies 
$$-\big[d'',[\Lambda,d']\big]+\big[d',[d'',\Lambda]\big]=0,$$
hence $\Delta''=\big[d',-\ii[d'',\Lambda]\big]=[d',d^{\prime\star}]=\Delta'$.
On the other hand
$$\Delta=[d'+d'',d^{\prime\star}+d^{\prime\prime\star}]=\Delta'+\Delta''+[d',d^{\prime\prime\star}]
+[d'',d^{\prime\star}].$$
Thus, it is enough to prove:
\endproof

\claim{(6.6) Lemma} $[d',d^{\prime\prime\star}]=0,~[d'',d^{\prime\star}]=0$.
\endclaim

\proof{Proof} We have $[d',d^{\prime\prime\star}]=-\ii\big[d',[\Lambda,d']\big]$ and 
(5.7) implies
$$-\big[d',[\Lambda,d']\big]+\big[\Lambda,[d',d']\big]+
\big[d',[d',\Lambda]\big]=0,$$
hence $-2\big[d',[\Lambda,d']\big]=0$ and $[d',d^{\prime\prime\star}]=0$. The second
relation 
$[d'',d^{\prime\star}]=0$ is the adjoint of the first.\qed
\endproof

\claim{(6.7) Theorem} $\Delta$ commutes with all operators 
$\star ,d',d'',d^{\prime\star},d^{\prime\prime\star},L,\Lambda$.
\endclaim

\proof{Proof} The identities $[d',\Delta']=[d^{\prime\star},\Delta']=0$,
$[d'',\Delta'']=[d^{\prime\prime\star},\Delta'']=0$ and $[\Delta,\star]=0$
are immediate. Furthermore, the equality $[d',L]=d'\omega=0$ together
with the Jacobi identity implies
$$[L,\Delta']=\big[L,[d',d^{\prime\star}]\big]=
-\big[d',[d^{\prime\star},L]\big]=\ii[d',d'']=0.$$
By adjunction, we also get $[\Delta',\Lambda]=0$.\qed
\endproof
 
\subsection{6.2}{Commutation Relations on Hermitian Manifolds}
We are going to extend the commutation relations of \S$\,$6.1 to an arbitrary
hermitian manifold $(X,\omega)$. In that case $\omega$ is no longer tangent
to a constant metric, and the commutation relations involve extra terms
arising from the {\it torsion} of $\omega$. Theorem~6.8 below is taken
from \cite{Demailly~1984}, but the idea was already contained in 
\cite{Griffiths~1966}.

\claim{(6.8) Theorem}  Let $\tau$ be the operator of type $(1,0)$ and order
$0$ defined by $\tau=[\Lambda, d'\omega]$. Then
\medskip\noindent
$\cmalign{
&{\rm a)}\qquad&[d^{\prime\prime\star},L]&=&\ii(d'+\tau),\cr
&{\rm b)}\qquad&[d^{\prime\star},L]&=-&\ii(d''+\ol\tau),\cr
&{\rm c)}\qquad&[\Lambda,d'']&=-&\ii(d^{\prime\star}+\tau^\star),\cr
&{\rm d)}\qquad&[\Lambda,d']&=&\ii(d^{\prime\prime\star}+\ol\tau^\star)~;\cr}$
\medskip\noindent
$d'\omega$ will be called the torsion form of $\omega$, 
and $\tau$ the torsion operator.
\endclaim

\proof{Proof} b) follows from a) by conjugation, whereas c), d) follow
from a), b)  by adjunction. It is therefore enough to prove relation a).

Let  $(z_j)_{1\le j\le n}$ be complex coordinates centered at a point
$x_0\in X$, such that the $n$-tuple
$(\partial/\partial z_1,\ldots,\partial/\partial z_n)$ 
is an orthonormal basis of  $T_{x_0}X$ for the metric
$\omega(x_0)$. Consider the metric with constant coefficients
$$\omega_0=\ii\sum_{1\le j\le n} dz_j\wedge d\ol z_j.$$
The metric $\omega$ can then be written
$$\omega=\omega_0+\gamma~~{\rm with}~~\gamma=O(|z|).$$
Denote by $\langle~,~\rangle_0,~L_0,~\Lambda_0,~d^{\prime\star}_0,~d^{\prime\prime\star}_0$
the inner product and the operators asso\-ciated to the constant metric
$\omega_0$, and let $dV_0=\omega^n_0/2^nn!$. The proof of relation~a)
is based on a Taylor expansion of $L,~\Lambda,~d^{\prime\star},~d^{\prime\prime\star}$
in terms of the operators with constant coefficients $L_0,~\Lambda_0,
~d^{\prime\star}_0,~d^{\prime\prime\star}_0$.
\endproof

\claim{(6.9) Lemma} Let $u,v\in \cC^\infty(X,\Lambda^{p,q}T^\star_X)$. Then in a
neighborhood of $x_0$
$$\langle u,v\rangle\,dV=\langle u-[\gamma,\Lambda_0]u,v\rangle_0\,dV_0
+O(|z|^2).$$
\endclaim

\proof{Proof} In a neighborhood of $x_0$, let
$$\gamma=\ii\sum_{1\le j\le n}\gamma_j\,\zeta^\star_j\wedge\ol\zeta^\star_j,
~~~~\gamma_1\le\gamma_2\le\cdots\le\gamma_n,$$
be a diagonalization of the (1,1)-form $\gamma(z)$ with respect to an
orthonormal basis $(\zeta_j)_{1\le j\le n}$ of $T_zX$ for $\omega_0(z)$.
We thus have
$$\omega=\omega_0+\gamma=\ii\sum\lambda_j\,\zeta^\star_j\wedge
\ol\zeta^\star_j$$
with $\lambda_j=1+\gamma_j$ and $\gamma_j=O(|z|)$. Set now
$$J=\{ j_1,\ldots,j_p\},\qquad\zeta^\star_J=\zeta^\star_{j_1}\wedge\cdots\wedge
\zeta^\star_{j_p},\qquad\lambda_J=\lambda_{j_1}\cdots\lambda_{j_p},$$
$$u=\sum u_{J,K}\,\zeta^\star_J\wedge\ol\zeta^\star_K,\qquad
v=\sum v_{J,K}\,\zeta^\star_J\wedge\ol\zeta^\star_K$$
where summations are extended to increasing multi-indices~ $J$, $K$~ such 
that 
$|J|=p$, $|K|=q$. With respect to  $\omega$ we have
$\langle\zeta^\star_j,\zeta^\star_j\rangle=\lambda^{-1}_j$,
hence
$$\eqalign{
\langle u,v\rangle\,dV&=\sum_{J,K}\lambda^{-1}_J\lambda^{-1}_K\,
u_{J,K}\ol v_{J,K}\,\lambda_1\cdots\lambda_n\, dV_0\cr
&=\sum_{J,K}\bigg(1-\sum_{j\in J}\gamma_j-\sum_{j\in K}\gamma_j
+\sum_{1\le j\le n}\gamma_j\bigg)
u_{J,K}\ol v_{J,K}\,dV_0+O(|z|^2).\cr}$$
Lemma 6.9 follows if we take Prop.~5.8 into account.\qed
\endproof

\claim{(6.10) Lemma} $d^{\prime\prime\star}=d^{\prime\prime\star}_0+
\big[\Lambda_0,[d^{\prime\prime\star}_0,\gamma]\big]$ at point $x_0$, i.e.\ at this
point both operators have the same formal expansion.
\endclaim

\proof{Proof} Since $d^{\prime\prime\star}$ is an operator of order 1, Lemma~6.9
shows that $d^{\prime\prime\star}$ coincides at point $x_0$ with the formal
adjoint of $d''$ for the metric
$$\Ll u,v\Gg_1=\int_X\langle u-[\gamma,\Lambda_0]u,v\rangle_0\,dV_0.$$
For any compactly supported  $u\in \cC^\infty(X,\Lambda^{p,q}T^\star_X)$,
$v\in \cC^\infty(X,\Lambda^{p,q-1}T^\star_X)$ we get by definition
$$\Ll u,d''v\Gg_1=\int_X\langle u-[\gamma,\Lambda_0]u,d''v\rangle_0\,dV_0
=\int_X\langle d^{\prime\prime\star}_0u-d^{\prime\prime\star}_0[\gamma,\Lambda_0]u,v\rangle_0\,dV_0.$$
Since $\omega$ and $\omega_0$ coincide at point $x_0$ and since $\gamma(x_0)=0$
we obtain at this point
$$\eqalign{
d^{\prime\prime\star}u&=d^{\prime\prime\star}_0u-d^{\prime\prime\star}_0[\gamma,\Lambda_0]u=d^{\prime\prime\star}_0u-\big[d^{\prime\prime\star}_0,[\gamma,\Lambda_0]\big]u~;\cr
d^{\prime\prime\star}&=d^{\prime\prime\star}_0-\big[d^{\prime\prime\star}_0,[\gamma,\Lambda_0]\big].\cr}$$
We have $[\Lambda_0,d^{\prime\prime\star}_0]=[d'',L_0]^\star=0$ since $d''\omega_0=0$.
The Jacobi identity (5.7) implies
$$\big[d^{\prime\prime\star}_0,[\gamma,\Lambda_0]\big]+\big[\Lambda_0,[d^{\prime\prime\star}_0,\gamma]
\big]=0,$$
and Lemma~6.10 follows.\qed
\endproof

\proof{Proof Proof of formula {\rm 6.8 a)}} The equality
$L=L_0+\gamma$ and Lemma~6.10 yield
$$[L,d^{\prime\prime\star}]=[L_0,d^{\prime\prime\star}_0]+\Big[L_0,\big[\Lambda_0,[d^{\prime\prime\star}_0,\gamma]
\big]\Big]+[\gamma,d^{\prime\prime\star}_0]\leqno(6.11)$$
at point $x_0$, because the triple bracket involving $\gamma$ twice vanishes
at~$x_0$. From the Jacobi identity applied to $C=[d^{\prime\prime\star}_0,\gamma]$, we get
$$\cases{\phantom{\Big(}
\big[L_0,[\Lambda_0,C]\big] 
=-[\Lambda_0,[C,L_0]\big]-\big[C,[L_0,\Lambda_0]\big],\cr
\phantom{\Big(}
[C,L_0]=\big[L_0,[d^{\prime\prime\star}_0,\gamma]\big]=\big[\gamma,[L_0,d^{\prime\prime\star}_0]\big]
~~~{\rm(since}~~[\gamma,L_0]=0).\cr}\leqno(6.12)$$
Lemma 6.3 yields $[L_0,d^{\prime\prime\star}_0]=-\ii d'$, hence
$$[C,L_0]=-[\gamma,\ii d']=\ii d'\gamma=\ii d'\omega .\leqno(6.13)$$
On the other hand, $C$ is of type $(1,0)$ and Cor.~5.9 gives
$$\big[ C,[L_0,\Lambda_0]\big]=-C=-[d^{\prime\prime\star}_0,\gamma].
\leqno(6.14)$$
From (6.12), (6.13), (6.14) we get
$$\Big[L_0,\big[\Lambda_0,[d^{\prime\prime\star}_0,\gamma]\big]\Big]=-[\Lambda_0,\ii d'\omega]+[d^{\prime\prime\star}_0,\gamma].$$
This last equality combined with (6.11) implies
$$[L,d^{\prime\prime\star}]=[L_0,d^{\prime\prime\star}_0]-[\Lambda_0,\ii d'\omega]=-\ii(d'+\tau)$$
at point $x_0$. Formula 6.8 a) is proved.\qed
\endproof

\claim{(6.15) Corollary} The complex Laplace-Beltrami operators satisfy
$$\eqalign{\Delta''&=\Delta'+[d',\tau^\star]-[d'',\ol\tau^\star],\cr
[d',d^{\prime\prime\star}]&=-[d',\ol\tau^\star],~~~~[d'',d^{\prime\star}]=-[d'',\tau^\star],\cr
\Delta&=\Delta'+\Delta''-[d',\ol\tau^\star]-[d'',\tau^\star].\cr}$$
Therefore $\Delta'$, $\Delta''$ and ${1\over 2}\Delta$ no longer
coincide, but they differ by linear differential operators of order 1 only.
\endclaim

\proof{Proof} As in the K\"ahler case (Cor.~6.5 and Lemma~6.6), 
we find
$$\eqalign{\Delta''
&=[d'',d^{\prime\prime\star}]=\big[d'',-i[\Lambda,d']-\ol\tau^\star]\cr
&=\big[d',-\ii[d'',\Lambda]\big]-[d'',\ol\tau^\star\big]=
\Delta'+[d',\tau^\star]-[d'',\ol\tau^\star],\cr
[d',d^{\prime\prime\star}+\ol\tau^\star]
&=-\ii\big[d',[\Lambda,d']\big]=0,\cr}$$
and the first two lines are proved. The third one is an immediate
consequence of the second.\qed
\endproof

\section{7}{Groups $\cH^{p,q}(X,E)$ and Serre Duality}
Let $(X,\omega)$ be a {\it compact hermitian} manifold and $E$ a holomorphic
hermitian vector bundle of rank $r$ over $X$. We denote by  $D_E$ the Chern
connection of $E$, by $D_E^\star=-\star D_E\,\star$ the formal adjoint of 
$D_E$, and by $D^{\prime\star}_E\,,~D^{\prime\prime\star}_E$ the components
of $D_E^\star$ of type $(-1,0)$ and $(0,-1)$.

Corollary 6.8 implies that the principal part of the operator
$\Delta''_E=D''D^{\prime\prime\star}_E+D^{\prime\prime\star}_E D''$ is one
half that of $\Delta_E$. The operator $\Delta''_E$ acting
on each space $\cC^\infty(X,\Lambda^{p,q}T^\star_X\otimes E)$
is thus a self-adjoint elliptic operator. Since  $D^{\prime\prime 2}=0$,
the following results can be obtained in a way similar to those of \S$\,$3.3.

\claim{(7.1) Theorem} For every bidegree $(p,q)$, there exists an orthogonal 
decomposition
$$\cC^\infty(X,\Lambda^{p,q}T^\star_X\otimes E)=\cH^{p,q}(X,E)\oplus\Im D''_E
\oplus\Im D^{\prime\prime\star}_E$$
where $\cH^{p,q}(X,E)$ is the space of $\Delta''_E$-harmonic forms in
$\cC^\infty(X,\Lambda^{p,q}T^\star_X\otimes E)$.
\endclaim

The above decomposition shows that the subspace of $d''$-cocycles in
$\cC^\infty(X,\Lambda^{p,q}T^\star_X\otimes E)$ is $\cH^{p,q}(X,E)\oplus
\Im D''_E$. From this, we infer

\claim{(7.2) Hodge isomorphism theorem} The Dolbeault cohomology group
$H^{p,q}(X,E)$ is finite dimensional, and there is an isomorphism
$$H^{p,q}(X,E)\simeq\cH^{p,q}(X,E).$$
\endclaim

\claim{(7.3) Serre duality theorem} The bilinear pairing
$$H^{p,q}(X,E)\times H^{n-p,n-q}(X,E^\star)\longrightarrow\bC,\qquad
(s,t)\longmapsto\int_M s\wedge t$$
is a non degenerate duality.
\endclaim

\proof{Proof} Let $s_1\in \cC^\infty(X,\Lambda^{p,q}T^\star_X\otimes E)$, $s_2\in \cC^\infty
(X,\Lambda^{n-p,n-q-1}T^\star_X\otimes E)$.
Since $s_1\wedge s_2$ is of bidegree $(n,n-1)$, we have
$$d(s_1\wedge s_2)=d''(s_1\wedge s_2)=d''s_1\wedge s_2+(-1)^{p+q} s_1
\wedge d''s_2.\leqno(7.4)$$
Stokes' formula implies that the above bilinear pairing can be
factorized through Dolbeault cohomology groups. The $\#$ operator defined
in \S$\,$3.1 is such that
$$\#:\cC^\infty(X,\Lambda^{p,q}T^\star_X\otimes E)\longrightarrow \cC^\infty(X,
\Lambda^{n-p,n-q}T^\star_X\otimes E^\star).$$
Furthermore, (3.20) implies
$$d''(\#\,s)=(-1)^{\deg s}\#\,D^{\prime\prime\star}_Es,~~~~
D^{\prime\prime\star}_{E^\star}(\#\,s)=(-1)^{\deg s+1}\#\,D^{\prime\prime\star}_Es,$$
$$\Delta''_{E^\star}(\#\,s)=\#\,\Delta''_Es,$$
where $D_{E^\star}$ is the Chern connection of $E^\star$. Consequently, 
$s\in\cH^{p,q}(X,E)$ if and only if
$\#\,s\in\cH^{n-p,n-q}(X,E^\star)$. Theorem~7.3
is then a consequence of the fact that the integral
$\|s\|^2=\int_X s\wedge\#\,s$ does not vanish unless $s=0$.\qed
\endproof

\section{8}{Cohomology of Compact K\"ahler Manifolds}
\subsection{8.1}{Bott-Chern Cohomology Groups}
Let $X$ be for the moment an arbitrary complex manifold. The following
``cohomology'' groups are helpful to describe Hodge theory on
compact complex manifolds which are not necessarily K\"ahler.

\claim{(8.1) Definition} We define the Bott-Chern cohomology groups
of $X$ to be
$$H_\BC^{p,q}(X,\bC)=\big(\cC^\infty(X,\Lambda^{p,q}T^\star_X)\cap\ker d\big)/
d'd''\cC^\infty(X,\Lambda^{p-1,q-1}T^\star_X).$$
Then $H_\BC^{\bu,\bu}(X,\bC)$ has the structure of a bigraded algebra,
which we call the Bott-Chern cohomology algebra of~$X$.
\endclaim

As the group $d'd''\cC^\infty(X,\Lambda^{p-1,q-1}T^\star_X)$ is contained in 
both coboundary groups\break
$d''\cC^\infty(X,\Lambda^{p,q-1}T^\star_X)$ or $d\,\cC^\infty(X,
\Lambda^{p+q-1}(\bC\otimes T_X)^\star)$, there are 
cano\-nical morphisms
$$\leqalignno{
&H_\BC^{p,q}(X,\bC)\longrightarrow H^{p,q}(X,\bC),&(8.2)\cr
&H_\BC^{p,q}(X,\bC)\longrightarrow H^{p+q}_\DR(X,\bC),&(8.3)\cr}$$
of the Bott-Chern cohomology to the Dolbeault or De Rham cohomology.
These morphisms are homomorphisms of $\bC$-algebras. 
It is also clear from the definition that we have the symmetry
property $H_\BC^{q,p}(X,\bC)=\ol{H_\BC^{p,q}(X,\bC)}$.
It can be shown from the Hodge-Fr\"olicher spectral sequence (see \S$\,$11
and Exercise 13.??) that $H_\BC^{p,q}(X,\bC)$ is always
finite dimensional if $X$ is compact.

\subsection{8.2}{Hodge Decomposition Theorem}
We suppose from now on that $(X,\omega)$ is a {\it compact K\"ahler}
manifold. The equality $\Delta=2\Delta''$ shows that $\Delta$ is homogeneous
with respect to bidegree and that there is an orthogonal decomposition
$$\cH^k(X,\bC)=\bigoplus_{p+q=k}\cH^{p,q}(X,\bC).\leqno(8.4)$$
As $\ol{\Delta''}=\Delta'=\Delta''$, we also have $\cH^{q,p}(X,\bC)=
\ol{\cH^{p,q}(X,\bC)}$. Using the Hodge isomorphism theorems
for the De Rham and Dolbeault cohomology, we get:

\claim{(8.5) Hodge decomposition theorem} On a compact K\"ahler manifold,
there are canonical isomorphisms
$$\matrix{\displaystyle
H^k(X,\bC)\simeq\bigoplus_{p+q=k}H^{p,q}(X,\bC)\hfill&\quad
\hbox{$($Hodge decomposition$)$},\hfill\cr
H^{q,p}(X,\bC)\simeq\ol{H^{p,q}(X,\bC)}\hfill&\quad
\hbox{$($Hodge symmetry$)$}.\hfill\cr}$$
\endclaim

The only point which is not a priori completely clear is that this
decomposition is independent of the K\"ahler metric. In order to show that
this is the case, one can use the following Lemma, which allows us to
compare all three types of cohomology groups considered in~\S$\,$8.1.

\claim{(8.6) Lemma} Let $u$ be a $d$-closed $(p,q)$-form. The following 
properties are equivalent:
{\parindent=7.2mm
\smallskip
\item{\swapleft{6pt}{${\rm a})$}} $u$ is $d$-exact$\,;$
\smallskip
\item{\swapleft{6pt}{${\rm b'})$}} $u$ is $d'$-exact$\,;$
\smallskip
\item{\swapleft{6pt}{${\rm b''})$}} $u$ is $d''$-exact$\,;$
\smallskip
\item{\swapleft{6pt}{${\rm c})$}} $u$ is $d'd''$-exact, i.e.\
$u$ can be written $u=d'd''v$.
\smallskip
\item{\swapleft{6pt}{${\rm d})$}} $u$ is orthogonal to $\cH^{p,q}(X,\bC)$.
\vskip0pt}
\endclaim

\proof{Proof} It is obvious that $c)$ implies a), ${\rm b')}$, ${\rm b'')}$ and
that a) or ${\rm b')}$ or ${\rm b'')}$ implies $d)$. It is thus sufficient
to prove that d) implies~c).  As $du=0$, we have $d'u=d''u=0$, and 
as $u$ is supposed to be orthogonal to $\cH^{p,q}(X,\bC)$,
Th.~7.1 implies $u=d''s$, $s\in \cC^\infty(X,\Lambda^{p,q-1}T^\star_X)$. By the
analogue of Th.~7.1 for $d'$, we have $s=h+d'v+d^{\prime\star}w$, with
$h\in\cH^{p,q-1}(X,\bC)$, $v\in \cC^\infty(X,\Lambda^{p-1,q-1}T^\star_X)$ and 
$w\in \cC^\infty(X,\Lambda^{p+1,q-1}T^\star_X)$. Therefore 
$$u=d''d'v+d''d^{\prime\star}w=-d'd''v-d^{\prime\star}d''w$$
in view of Lemma~6.6. As $d'u=0$, the component $d^{\prime\star}d''w$
orthogonal to $\ker d'$ must be zero.\qed
\endproof

From Lemma~8.6 we infer the following Corollary, which in turn implies
that the Hodge decomposition does not depend on the K\"ahler metric.

\claim{(8.7) Corollary} Let $X$ be a compact K\"ahler manifold. Then
the natural morphisms
$$H^{p,q}_\BC(X,\bC)\longrightarrow H^{p,q}(X,\bC),\qquad
\bigoplus_{p+q=k}H^{p,q}_\BC(X,\bC)\longrightarrow H^k_\DR(X,\bC)$$
are isomorphisms.
\endclaim

\proof{Proof} The surjectivity of $H^{p,q}_\BC(X,\bC)\to H^{p,q}(X,\bC)$
comes from the fact that every class in $H^{p,q}(X,\bC)$ can be represented
by a harmonic $(p,q)$-form, thus by a $d$-closed $(p,q)$-form; the
injectivity means nothing more than the equivalence (8.5$\,{\rm b}'')
\Leftrightarrow(8.5\,$c). Hence $H^{p,q}_\BC(X,\bC)\simeq H^{p,q}(X,\bC)
\simeq\cH^{p,q}(X,\bC)$, and the isomorphism
$\bigoplus_{p+q=k}H^{p,q}_\BC(X,\bC)\longrightarrow H^k_\DR(X,\bC)$
follows from (8.4).\qed
\endproof

Let us quote now two simple applications of Hodge theory. The first of these
is a computation of the Dolbeault cohomology groups of $\bP^n$.
As $H^{2p}_\DR(\bP^n,\bC)=\bC$ and
$H^{p,p}(\bP^n,\bC)\ni\{\omega^p\}\ne 0$, the Hodge decomposition
formula implies:

\claim{(8.8) Application} The Dolbeault cohomology groups of $\bP^n$ are 
$$H^{p,p}(\bP^n,\bC)=\bC\quad\hbox{for}~0\le p\le n,\qquad
H^{p,q}(\bP^n,\bC)=0\quad\hbox{for}~p\ne q.\eqno\square$$
\endclaim

\claim{(8.9) Proposition} Every holomorphic $p$-form on a
compact K\"ahler mani\-fold $X$ is $d$-closed.
\endclaim

\proof{Proof} If $u$ is a holomorphic form of type $(p,0)$ then
$d''u=0$. Furthermore $d^{\prime\prime\star}u$ is of type $(p,-1)$, hence
$d^{\prime\prime\star}u=0$. Therefore $\Delta u=2\Delta''u=0$, which implies 
$du=0$.\qed
\endproof

\claim{(8.10) Example} \rm Consider the {\it Heisenberg group}
$G\subset{\rm Gl}_3(\bC)$, defined as the subgroup of matrices
$$M=\pmatrix{
1~~&\hfill x~~&\hfill z\hfill\cr
0~~&\hfill 1~~&\hfill y\hfill\cr
0~~&\hfill 0~~&\hfill 1\hfill\cr}\quad,\quad (x,y,z)\in\bC^3.$$
Let $\Gamma$ be the discrete subgroup of matrices with entries 
$x,y,z\in\bZ[\ii]$ (or more generally in the ring of integers of an
imaginary quadratic field). Then $X=G/\Gamma$ is a compact complex
3-fold, known as the {\it Iwasawa manifold}. The equality
$$M^{-1}dM=\pmatrix{0&dx&dz-xdy\cr 0&0&dy\cr 0&0&0\cr}$$
shows that $dx,~dy,~dz-xdy$ are left invariant holomorphic $1$-forms on
$G$. These forms induce holomorphic $1$-forms on the quotient $X=G/\Gamma$.
Since $dz-xdy$ is not $d$-closed, we see that $X$ cannot be K\"ahler.
\endclaim

\subsection{8.3}{Primitive Decomposition and Hard Lefschetz Theorem}
We first introduce some standard notation. The {\it Betti numbers} and
{\it Hodge numbers} of $X$ are by definition
$$b_k=\dim_\bC H^k(X,\bC),~~~~h^{p,q}=\dim_\bC H^{p,q}(X,\bC).
\leqno(8.11)$$
Thanks to Hodge decomposition, these numbers satisfy the relations
$$b_k=\sum_{p+q=k} h^{p,q},~~~~h^{q,p}=h^{p,q}.\leqno(8.12)$$
As a consequence, the Betti numbers $b_{2k+1}$ of a compact K\"ahler manifold
are even. Note that the Serre duality theorem gives the additional
relation \hbox{$h^{p,q}=h^{n-p,n-q}$}, which holds as soon as $X$ is
compact. The existence of primitive decomposition implies other interesting
specific features of the cohomology algebra of compact K\"ahler manifolds.

\claim{(8.13) Lemma} If $u=\sum_{r\ge (k-n)_+}L^ru_r$ is the primitive
decomposition of a harmonic $k$-form $u$, then all components
$u_r$ are harmonic.
\endclaim

\proof{Proof} Since  $[\Delta,L]=0$, we get $0=\Delta u=\sum_r L^r\Delta
u_r$, hence $\Delta u_r=0$ by uniqueness.\qed
\endproof

Let us denote by $\Prim\cH^k(X,\bC)=\bigoplus_{p+q=k}\Prim\cH^{p,q}(X,\bC)$
the spaces of primitive harmonic $k$-forms and let $b_{k,\prim}$,
$h_\prim^{p,q}$ be 
their respective dimensions. Lemma 8.13 yields
$$\leqalignno{
\cH^{p,q}(X,\bC)&=\bigoplus_{r\ge (p+q-n)_+}L^r\Prim\cH^{p-r,q-r}(X,\bC),
&(8.14)\cr
h^{p,q}&=\sum_{r\ge (p+q-n)_+}h_\prim^{p-r,q-r}.&(8.15)\cr}$$
Formula (8.15) can be rewritten
$$\cases{\phantom{\Big(}
{\rm if}~~p+q\le n,~~h^{p,q}=h_\prim^{p,q}+h_\prim^{p-1,q-1}+\cdots\cr
\phantom{\Big(}
{\rm if}~~p+q\ge n,~~h^{p,q}=h_\prim^{n-q,n-p}+h_\prim^{n-q-1,n-p-1}+\cdots.
\cr}\leqno(8.15')$$

\claim{(8.16) Corollary} The Hodge and Betti numbers satisfy the inequalities
\smallskip
\item{\rm a)} if $k=p+q\le n$,
then~ $h^{p,q}\ge h^{p-1,q-1},~~b_k\ge b_{k-2}$,
\smallskip
\item{\rm b)} if $k=p+q\ge n$,
then~ $h^{p,q}\ge h^{p+1,q+1},~~b_k\ge b_{k+2}$.\qed
\endclaim

Another important result of Hodge theory (which is in fact a direct
consequence of Cor.~5.17) is the

\claim{(8.17) Hard Lefschetz theorem} The mappings
$$\cmalign{
&L^{n-k}~:~~\hfill H^k(X,\bC)&\longrightarrow H^{2n-k}(X,\bC),~~~~&k\le n,\cr
&L^{n-p-q}~:~~H^{p,q}(X,\bC)&\longrightarrow H^{n-q,n-p}(X,\bC),~~~~&p+q\le n,\cr}$$
are isomorphisms.\qed
\endclaim

\section{9}{Jacobian and Albanese Varieties}
\subsection{9.1}{Description of the Picard Group}
An important application of Hodge theory is a description of the Picard 
group $H^1(X,\cO^\star)$ of a compact K\"ahler manifold. We assume here that
$X$ is connected. The exponential exact
sequence $0\to\bZ\to\cO\to\cO^\star\to 1$ gives
$$\leqalignno{
0\longrightarrow&H^1(X,\bZ)\longrightarrow H^1(X,\cO)\longrightarrow H^1(X,\cO^\star)&(9.1)\cr
\buildo c_1\over\longrightarrow&H^2(X,\bZ)\longrightarrow H^2(X,\cO)\cr}$$
because the map $\exp(2\pi i\bu):H^0(X,\cO)=\bC\longrightarrow H^0(X,\cO^\star)=\bC^\star$
is onto. 
We have $H^1(X,\cO)\simeq H^{0,1}(X,\bC)$ by (V-11.6). The dimension
of this group is called the {\it irregularity of $X$} and is usually denoted
$$q=q(X)=h^{0,1}=h^{1,0}.\leqno(9.2)$$
Therefore we have $b_1=2q$ and
$$H^1(X,\cO)\simeq\bC^q,~~~~H^0(X,\Omega^1_X)=H^{1,0}(X,\bC)
\simeq\bC^q.\leqno(9.3)$$

\claim{(9.4) Lemma} The image of $H^1(X,\bZ)$ in $H^1(X,\cO)$ is a lattice.
\endclaim

\proof{Proof} Consider the morphisms
$$H^1(X,\bZ)\longrightarrow H^1(X,\bR)\longrightarrow H^1(X,\bC)\longrightarrow H^1(X,\cO)$$
induced by the inclusions $\bZ\subset\bR\subset\bC\subset\cO$. Since the
\v Cech cohomology groups with values in $\bZ$, $\bR$ can be computed
by finite acyclic coverings, we see that $H^1(X,\bZ)$ is a finitely 
generated $\bZ$-module and that the image of $H^1(X,\bZ)$ in $H^1(X,\bR)$
is a lattice. It is enough to check that the map $H^1(X,\bR)\longrightarrow H^1(X,\cO)$
is an isomorphism. However, the commutative \hbox{diagram}
$$\cmalign{
&0&\longrightarrow&\bC&\longrightarrow&~\cE^0&\buildo d\over\longrightarrow&~\cE^1&\buildo d\over\longrightarrow&~\cE^2
&\longrightarrow&\cdots\cr
&&&\big\downarrow&&~\big\downarrow&&~\big\downarrow&&~\big\downarrow&&\cr
&0&\longrightarrow&\cO&\longrightarrow&\cE^{0,0}&\buildo d''\over\longrightarrow&\cE^{0,1}&\buildo d''\over\longrightarrow&
\cE^{0,2}&\longrightarrow&\cdots\cr}$$
shows that the map $H^1(X,\bR)\longrightarrow H^1(X,\cO)$ corresponds in De Rham and
Dolbeault cohomologies to the composite mapping
$$H^1_{DR}(X,\bR)\subset H^1_{DR}(X,\bC)\longrightarrow H^{0,1}(X,\bC).$$
Since~ $H^{1,0}(X,\bC)$ and~ $H^{0,1}(X,\bC)$ are complex conjugate
subspaces in
$H^1_{DR}(X,\bC)$, we conclude that $H^1_{DR}(X,\bR)\longrightarrow H^{0,1}(X,\bC)$
is an isomorphism.\qed
\endproof

As a consequence of this lemma, $H^1(X,\bZ)\simeq\bZ^{2q}$.
The $q$-dimensional complex torus
$$\Jac(X)=H^1(X,\cO)/H^1(X,\bZ)\leqno(9.5)$$
is called the {\it Jacobian variety of $X$} and is isomorphic to the
subgroup of $H^1(X,\cO^\star)$ corresponding to line bundles of zero
first Chern class. On the other hand, the kernel of
$$H^2(X,\bZ)\longrightarrow H^2(X,\cO)=H^{0,2}(X,\bC)$$
which consists of integral cohomology classes of type~$(1,1)$,
is equal to the image of $c_1$ in $H^2(X,\bZ)$. This subgroup is called
the {\it Neron-Severi group} of~$X$, and is denoted $NS(X)$. The exact 
sequence (9.1) yields
$$0\longrightarrow\Jac(X)\longrightarrow H^1(X,\cO^\star)\buildo c_1\over\longrightarrow NS(X)\longrightarrow 0.\leqno(9.6)$$
The Picard group $H^1(X,\cO^\star)$ is thus an extension of the complex torus
$\Jac(X)$ by the finitely generated $\bZ$-module $NS(X)$.

\claim{(9.7) Corollary} The Picard group of $\bP^n$ is $H^1(\bP^n,\cO^\star)
\simeq\bZ$, and every line bundle over $\bP^n$ is isomorphic to one of the 
line bundles $\cO(k)$, $k\in\bZ$.
\endclaim

\proof{Proof} We have $H^k(\bP^n,\cO)=H^{0,k}(\bP^n,\bC)=0$ for
$k\ge 1$ by Appl.~8.8, thus $\Jac({\bP^n})=0$ and
$NS(\bP^n)=H^2(\bP^n,\bZ)\simeq\bZ$. Moreover,
$c_1\big(\cO(1)\big)$ is a generator of $H^2(\bP^n,\bZ)$ in virtue 
of Th.~V-15.10.\qed
\endproof

\subsection{9.2}{Albanese Variety}
A proof similar to that of Lemma~9.4 shows that the image of 
$H^{2n-1}(X,\bZ)$ in $H^{n-1,n}(X,\bC)$ via the composite map
$$H^{2n-1}(X,\bZ)\to H^{2n-1}(X,\bR)\to H^{2n-1}(X,\bC)\to
H^{n-1,n}(X,\bC)\leqno(9.8)$$
is a lattice. The $q$-dimensional complex torus
$$\Alb(X)=H^{n-1,n}(X,\bC)/\Im H^{2n-1}(X,\bZ)\leqno(9.9)$$
is called the {\it Albanese variety of $X$}. We first give a slightly
different description of $\Alb(X)$, based on the Serre duality
isomorphism
$$H^{n-1,n}(X,\bC)\simeq\big(H^{1,0}(X,\bC)\big)^\star
\simeq\big(H^0(X,\Omega^1_X)\big)^\star.$$

\claim{(9.10) Lemma} For any compact oriented differentiable
manifold $M$ with $\dim_\bR M=m$, there is a natural isomorphism
$$H_1(M,\bZ)\to H^{m-1}(M,\bZ)$$
where $H_1(M,\bZ)$ is the first homology group of~$M$, that is, the
abelianization of~$\pi_1(M)$.
\endclaim

\proof{Proof} This is a well known consequence of Poincar\'e duality,
see e.g.\ \cite{Spanier 1966}. We will content ourselves with a description
of the morphism. Fix a base point $a\in M$. Every homotopy class
$[\gamma]\in\pi_1(M,a)$ can be represented by as a composition of
closed loops diffeomorphic to~$S^1$. Let~$\gamma$ be such a loop.
As every oriented vector bundle over
$S^1$ is trivial, the normal bundle to $\gamma$ is trivial. Hence
$\gamma(S^1)$ has a neighborhood $U$ diffeomorphic to $S^1\times\bR^{m-1}$,
and there is a diffeomorphism $\varphi:S^1\times\bR^{m-1}\to U$ with 
$\varphi_{\restriction S^1\times\{0\}}=\gamma$. Let 
$\{\delta_0\}\in H^{m-1}_c(\bR^{m-1},\bZ)$ be the fundamental class 
represented by the Dirac measure $\delta_0\in\cD'_0(\bR^{m-1})$
in De Rham cohomology. Then the cartesian product $1\times\{\delta_0\}\in
H^{m-1}_c(S^1\times\bR^{m-1},\bZ)$ is 
represented by the current \hbox{$[S^1]\otimes\{\delta_0\}\in\cD'_1(S^1\times
\bR^{m-1})$} and the current of integration over~$\gamma$ is precisely
the direct image current
$$I_\gamma:=\varphi_\star([S^1]\otimes\delta_0)=(\varphi^{-1})^\star
([S^1]\otimes\delta_0).$$
Its cohomology class $\{I_\gamma\}\in H^{m-1}_c(U,\bR)$ is thus the
image of the class 
$$
(\varphi^{-1})^\star\big(1\times\{\delta_0\}\big)
\in H^{m-1}_c(U,\bZ).
$$
Therefore, we have obtained a well defined morphism
$$
\pi_1(M,a)\longrightarrow H^{m-1}_c(U,\bZ)\longrightarrow H^{m-1}(M,\bZ),
~~~~[\gamma]
\longmapsto(\varphi^{-1})^\star\big(1\times\{\delta_0\}\big)
$$
and the image of $[\gamma]$ in $H^{m-1}(M,\bR)$ is the De Rham
cohomology class of the integration current $I_\gamma$.\qed
\endproof

Thanks to Lemma~9.10, we can reformulate the definition of the
Albanese variety as
$$\Alb(X)=\big(H^0(X,\Omega^1_X)\big)^\star/\Im H_1(X,\bZ)\leqno(9.11)$$
where $H_1(X,\bZ)$ is mapped to $\big(H^0(X,\Omega^1_X)\big)^\star$ by
$$[\gamma]\longmapsto\wt I_\gamma=\Big(u\mapsto\int_\gamma u\Big).$$
Observe that the integral only depends on the homotopy class $[\gamma]$
because all holomorphic $1$-forms $u$ on $X$ are closed by Prop.~8.9.

We are going to show that there exists a canonical holomorphic map 
$\alpha:X\to\Alb(X)$. Let $a$ be a base point in $X$. For any
$x\in X$, we select a path $\xi$ from $a$ to $x$ and associate to
$x$ the linear form in $\big(H^0(X,\Omega^1_X)\big)^\star$ defined
by~$\wt I_\xi$. By construction the class of this linear form
mod $\Im H_1(X,\bZ)$ does not depend
on~$\xi$, since $\wt I_{\xi^{\prime\,-1}\xi}$ is in the
image of $H_1(X,\bZ)$ for any other path~$\xi'$. It is thus
legitimate to define the {\it Albanese map} as
$$\alpha:X\longrightarrow\Alb(X),~~~~x\longmapsto
\Big(u\mapsto\int_a^xu\Big)~~\hbox{\rm mod}~~\Im\,H_1(X,\bZ).
\leqno(9.12)$$
Of course, if we fix a basis $(u_1,\ldots,u_q)$ of $H^0(X,\Omega^1_X)$,
the Albanese map can be seen in coordinates as the map
$$\alpha:X\longrightarrow\bC^q/\Lambda,~~~~x\longmapsto
\Big(\int_a^xu_1,\ldots,\int_a^xu_q\Big)~~\hbox{\rm mod}~~\Lambda,
\leqno(9.13)$$
where $\Lambda\subset\bC^q$ is the {\it group of periods} of $(u_1,\ldots,u_q)\,$:
$$\Lambda=\Big\{\Big(\int_\gamma u_1,\ldots,\int_\gamma u_q\Big)~;~
[\gamma]\in\pi_1(X,a)\,\Big\}.\leqno(9.13')$$
It is then clear that $\alpha$ is a holomorphic map. With the
original definition (9.9) of the Albanese variety, it is not
difficult to see that $\alpha$ is the map given by
$$\alpha:X\longrightarrow\Alb(X),~~~~x\longmapsto\{I^{n-1,n}_\xi\}~~{\rm mod}~~
H^{2n-1}(X,\bZ),\leqno(9.14)$$
where $\{I^{n-1,n}_\xi\}\in H^{n-1,n}(X,\bC)$ denotes the
$(n-1,n)$-component of the De Rham cohomology class $\{I_\xi\}$.

\section{10}{Complex Curves}
We show here how Hodge theory can be used to derive quickly the basic properties
of compact manifolds of complex dimension $1$ (also called {\it complex
curves} or {\it Riemann surfaces}). Let $X$ be such a curve. We shall always 
assume in this section that $X$ is compact and connected. Since every positive 
$(1,1)$-form on a curve defines a K\"ahler metric, the results of \S$\,$8
and \S$\,$9 can be applied.

\subsection{10.1}{Riemann-Roch Formula}
Denoting $g=h^1(X,\cO)$, we find
$$\leqalignno{
&H^1(X,\cO)\simeq\bC^g,~~~~H^0(X,\Omega^1_X)\simeq\bC^g,
&(10.1)\cr
&H^0(X,\bZ)=\bZ,~~~~H^1(X,\bZ)=\bZ^{2g},~~~~H^2(X,\bZ)=\bZ.&(10.2)}$$
The classification of oriented topological surfaces shows that $X$
is homeomorphic to a sphere with $g$ handles ( $=$ torus
with $g$ holes), but this property will not be used in the sequel.
The number $g$ is called the {\it genus} of~$X$.

Any divisor on $X$ can be written $\Delta=\sum m_ja_j$ where $(a_j)$ is a
finite sequence of points and $m_j\in\bZ$. Let $E$ be a line bundle
over $X$. We shall identify $E$ and the associated locally free sheaf 
$\cO(E)$. According to V-13.2, we denote by $E(\Delta)$ the sheaf of germs of
meromorphic sections $f$ of $E$ such that ${\rm div}\,f+\Delta\ge 0$, 
i.e.\ which have a pole of order $\le m_j$ at $a_j$ if $m_j>0$, and which 
have a zero of order $\ge|m_j|$ at $a_j$ if $m_j<0$. Clearly
$$E(\Delta)=E\otimes\cO(\Delta),~~~~\cO(\Delta+\Delta')=\cO(\Delta)\otimes
\cO(\Delta').\leqno(10.3)$$
For any point $a\in X$ and any integer $m>0$, there is an exact sequence
$$0\longrightarrow E\longrightarrow E(m[a])\longrightarrow\cS\longrightarrow 0$$
where $\cS=E(m[a])/E$ is a sheaf with only one non zero stalk $\cS_a$
isomorphic to $\bC^m$. Indeed, if $z$ is a holomorphic coordinate near
$a$, the stalk $\cS_a$ corresponds to the polar parts
$\sum_{-m\le k<0} c_kz^k$ in the power series expansions of germs
of meromorphic sections at point $a$. We get an exact sequence
$$H^0\big(X,E(m[a])\big)\longrightarrow\bC^m\longrightarrow H^1(X,E).$$
When $m$ is chosen larger than $\dim H^1(X,E)$, we see that
$E(m[a])$ has a non zero section and conclude:

\claim{(10.4) Theorem} Let $a$ be a given point on a curve. Then every line 
bundle $E$ has non zero meromorphic sections $f$ with a pole at $a$ and
no other poles.
\endclaim

If $\Delta$ is the divisor of a meromorphic section $f$ of $E$, we have
$E\simeq\cO(\Delta)$, so the map 
$${\rm Div}(X)\longrightarrow H^1(X,\cO^\star),~~~~\Delta\longmapsto\cO(\Delta)$$ 
is onto (cf.\ (V-13.8)). On the other hand,
${\rm Div}$ is clearly a soft sheaf, thus $H^1(X,{\rm Div})=0$.
The long cohomology sequence associated to the exact sequence
$1\to\cO^\star\to\cM^\star\to{\rm Div}\to 0$ implies:

\claim{(10.5) Corollary} On any complex curve, one has $H^1(X,\cM^\star)=0$
and there is an exact sequence
$$0\longrightarrow\bC^\star\longrightarrow\cM^\star(X)\longrightarrow{\rm Div}(X)\longrightarrow H^1(X,\cO^\star)\longrightarrow 0.$$
\endclaim

The first Chern class $c_1(E)\in H^2(X,\bZ)$ can be interpreted
as an integer. This integer is called the {\it degree} of~$E$.
If $E\simeq\cO(\Delta)$ with $\Delta=\sum m_ja_j$, formula V-13.6
shows that the image of $c_1(E)$ in $H^2(X,\bR)$ is the De Rham
cohomology class of the associated current $[\Delta]=\sum m_j\delta_{a_j}$,
hence
$$c_1(E)=\int_X[\Delta]=\sum m_j.\leqno(10.6)$$
If $\sum m_ja_j$ is the divisor of a meromorphic
function, we have $\sum m_j=0$ because the associated bundle $E=\cO(\sum
m_ja_j)$ is trivial.

\claim{(10.7) Theorem} Let $E$ be a line bundle on a complex curve~$X$. Then
\smallskip
\item{\rm a)} $H^0(X,E)=0$~~if~~$c_1(E)<0$ or if~~$c_1(E)=0$ and
$E$ is non trivial$\,;$
\smallskip
\item{\rm b)} For every positive $(1,1)$-form $\omega$ on $X$ with
$\int_X\omega=1$, $E$ has a hermitian metric such that ${\ii\over 2\pi}
\Theta(E)=c_1(E)\,\omega$.  In particular, $E$ has a metric of positive
$($resp.\  negative$)$ curvature if and only if $c_1(E)>0$ $($resp.\ 
if and only if $c_1(E)<0)$.\smallskip
\endclaim

\proof{Proof} a) If $E$ has a non zero holomorphic section $f$, then its degree
is $c_1(E)=\int_X{\rm div}\,f\ge 0$. In fact, we even have $c_1(E)>0$
unless $f$ does not vanish, in which case $E$ is trivial.

b) Select an arbitrary hermitian metric $h$ on $E$. Then
$c_1(E)\,\omega-{\ii\over 2\pi}\Theta_h(E)$ is a real $(1,1)$-form
cohomologous to zero (the integral over $X$ is zero), so Lemma~8.6~c) implies
$$c_1(E)\,\omega-{\ii\over 2\pi}\Theta_h(E)=\ii d'd''\varphi$$
for some real function $\varphi\in \cC^\infty(X,\bR)$. If we replace the initial
metric of $E$ by $h'=h\,e^{-\varphi}$, we get a metric of constant
curvature $c_1(E)\,\omega$.\qed
\endproof
          
\claim{(10.8) Riemann-Roch formula} Let $E$ be a holomorphic
line bundle and let $h^q(E)=\dim H^q(X,E)$. Then
$$h^0(E)-h^1(E)=c_1(E)-g+1.$$
Moreover $h^1(E)=h^0(K\otimes E^\star)$, where $K=\Omega^1_X$ is
the canonical line bundle of~$X$.
\endclaim

\proof{Proof} We claim that for every line bundle $F$ and every divisor $\Delta$ 
we have the equality
$$h^0\big(F(\Delta)\big)-h^1\big(F(\Delta)\big)=h^0(F)-h^1(F)+\int_X[\Delta].
\leqno(10.9)$$
If we write $E=\cO(\Delta)$ and apply the above equality with $F=\cO$, the
Riemann-Roch formula results from (10.6), (10.9) and from the equalities
$$h^0(\cO)=\dim H^0(X,\cO)=1,~~~~h^1(\cO)=\dim H^1(X,\cO)=g.$$
However, (10.9) need only be proved when $\Delta\ge 0\,$: otherwise we
are reduced to this case by writing $\Delta=\Delta_1-\Delta_2$ with
$\Delta_1,\Delta_2\ge 0$ and by applying the result to the pairs
$(F,\Delta_1)$ and $\big(F(\Delta),\Delta_2\big)$. If $\Delta=\sum m_ja_j
\ge 0$, there is an exact sequence
$$0\longrightarrow F\longrightarrow F(\Delta)\longrightarrow\cS\longrightarrow 0$$
where $\cS_{a_j}\simeq\bC^{m_j}$ and the other stalks are zero. 
Let $m=\sum m_j=\int_X[\Delta]$. The sheaf $\cS$ is acyclic, because
its support $\{a_j\}$ is of dimension $0$. Hence there is
an exact sequence
$$0\longrightarrow H^0(F)\longrightarrow H^0\big(F(\Delta)\big)\longrightarrow\bC^m\longrightarrow H^1(F)\longrightarrow
H^1\big(F(\Delta)\big)\longrightarrow 0$$
and (10.9) follows. The equality $h^1(E)=h^0(K\otimes E^\star)$ is a
consequence of the Serre duality theorem
$$\big(H^{0,1}(X,E)\big)^\star\simeq H^{1,0}(X,E^\star),~~~~\hbox{\rm i.e.}~~
\big(H^1(X,E)\big)^\star\simeq H^0(X,K\otimes E^\star).\eqno{\square}$$
\endproof

\claim{(10.10) Corollary (Hurwitz' formula)} $c_1(K)=2g-2$.
\endclaim

\proof{Proof} Apply Riemann-Roch to $E=K$ and observe that
$$\left.\eqalign{
h^0(K)&=\dim H^0(X,\Omega^1_X)=g\cr
h^1(K)&=\dim H^1(X,\Omega^1_X)=h^{1,1}=b_2=1\cr}\right.\leqno(10.11)$$
\endproof

\claim{(10.12) Corollary} For every $a\in X$ and every $m\in\bZ$
$$h^0\big(K(-m[a])\big)=h^1\big(\cO(m[a])\big)=h^0\big(\cO(m[a])\big)-m+g-1.$$
\endclaim

\subsection{10.2}{Jacobian of a Curve}
By the Neron-Severi sequence (9.6), there is an exact sequence
$$0\longrightarrow\Jac(X)\longrightarrow H^1(X,\cO^\star)\buildo c_1\over\longrightarrow\bZ\longrightarrow 0,
\leqno(10.13)$$
where the Jacobian $\Jac(X)$ is a $g$-dimensional torus. Choose a base point
$a\in X$. For every point $x\in X$, the line bundle $\cO([x]-[a])$
has zero first Chern class, so we have a well-defined map
$$\Phi_a:X\longrightarrow\Jac(X),~~~~x\longmapsto\cO([x]-[a]).\leqno(10.14)$$
Observe that the Jacobian $\Jac(X)$ of a curve coincides by definition
with the Albanese variety $\Alb(X)$.

\claim{(10.15) Lemma} The above map $\Phi_a$ coincides with the Albanese
map \hbox{$\alpha:X\longrightarrow\Alb(X)$} defined in $(9.12)$.
\endclaim

\proof{Proof} By holomorphic continuation, it is enough to prove that
\hbox{$\Phi_a(x)=\alpha(x)$} when $x$ is near $a$. Let $z$ be a complex 
coordinate and let $D'\subset\!\subset D$ be open disks centered at~$a$.
Relatively to the covering
$$U_1=D,~~~~U_2=X\setminus\ol{D'},$$
the line bundle $\cO([x]-[a])$ is defined by the \v Cech cocycle $c\in 
C^1(\cU,\cO^\star)$ such that
$$c_{12}(z)={z-x\over z-a}~~~{\rm on}~~U_{12}=D\setminus\ol{D'}.$$
On the other hand, we compute $\alpha(x)$ by Formula~(9.14). The path
integral current $I_{[a,x]}\in\cD'_1(X)$ is equal to $0$ on~$U_2$.
Lemma~I-2.10 implies \hbox{$d''(dz/2\pi\ii z)=\delta_0\,d\ol z
\wedge dz/2i=\delta_0$} according to the usual identification of
distributions and currents of degree $0$, thus
$$I^{0,1}_{[a,x]}=d''\Big({dz\over2\pi iz}\star I^{0,1}_{[a,x]}\Big)~~~
{\rm on}~~U_1.$$
Therefore $\{I^{0,1}_{[a,x]}\}\in H^{0,1}(X,\bC)$ is equal to the \v Cech
cohomology class $\{c'\}$ in $H^1(X,\cO)$ represented by the cocycle
$$c'_{12}(z)={dw\over2\pi iw}\star I^{0,1}_{[a,x]}={1\over 2\pi i}\int_a^x 
{dw\over w-z}={1\over 2\pi\ii}\log{z-x\over z-a}~~~{\rm on}~~U_{12}$$
and we have $c=\exp(2\pi\ii c')$ in $H^1(X,\cO^\star)$.\qed
\endproof

The nature of $\Phi_a$ depends on the value of the genus $g$. A careful
examination of $\Phi_a$ will enable us to determine all curves of genus $0$
and $1$.  

\claim{(10.16) Theorem} The following properties are equivalent:
\smallskip
\item{\rm a)} $g=0\,;$
\smallskip
\item{\rm b)} $X$ has a meromorphic function $f$ having only one 
simple pole $p\,;$
\smallskip
\item{\rm c)} $X$ is biholomorphic to $\bP^1$.
\endclaim

\proof{Proof} c) $\Longrightarrow$ a) is clear.
\medskip
\noindent a) $\Longrightarrow$ b). Since $g=0$, we have $\Jac(X)=0$. If
$p,p'\in X$ are distinct points, the bundle $\cO([p']-[p])$ has zero first
Chern class, therefore it is trivial and there exists a meromorphic function
$f$ with ${\rm div}\,f=[p']-[p]$. In particular $p$ is the only pole
of $f$, and this pole is simple.
\medskip
\noindent b) $\Longrightarrow$ c). We may consider $f$ as a map $X\longrightarrow\bP^1=
\bC\cup\{\infty\}$. For every value $w\in\bC$, the function $f-w$ must have
exactly one simple zero  $x\in X$ because $\int_X{\rm div}(f-w)=0$ and $p$ is 
a simple pole. Therefore $f:X\to\bP^1$ is bijective and $X$ is biholomorphic
to $\bP^1$.\qed
\endproof

\claim{(10.17) Theorem} The map $\Phi_a$ is always injective for $g\ge 1$. 
\smallskip
\item{\rm a)} If $g=1$, $\Phi_a$ is a biholomorphism. In particular 
every curve of genus $1$ is biholomorphic to a complex torus $\bC/\Gamma$. 
\smallskip
\item{\rm b)} If $g\ge 2$, $\Phi_a$ is an embedding.
\endclaim

\proof{Proof} If $\Phi_a$ is not injective, there exist points $x_1\ne x_2$ such 
that $\cO([x_1]-[x_2])$ is trivial; then there is
a meromorphic function $f$ such that ${\rm div}\,f=[x_1]-[x_2]$
and Th.~10.16 implies that $g=0$.

When $g=1$, $\Phi_a$ is an injective map $X\longrightarrow\Jac(X)\simeq\bC/\Gamma$,
thus $\Phi_a$ is open. It follows that $\Phi_a(X)$ is a compact open subset
of $\bC/\Gamma$, so $\Phi_a(X)=\bC/\Gamma$ and $\Phi_a$ is a
biholomorphism of $X$ onto $\bC/\Gamma$.

In order to prove that $\Phi_a$ is an embedding when $g\ge 2$, it is
sufficient to show that the holomorphic $1$-forms $u_1,\ldots,u_g$ do not all
vanish at a given point $x\in X$. In fact, $X$ has no non constant
meromorphic function having only a simple pole at $x$, thus 
$h^0\big(\cO([x])\big)=1$ and Cor.~10.12 implies
$$h^0\big(K(-[x])\big)=g-1<h^0(K)=g.$$
Hence $K$ has a section $u$ which does not vanish at $x$.\qed
\endproof

\subsection{10.3}{Weierstrass Points of a Curve}
We want to study how many meromorphic functions have a unique pole of
multiplicity $\le m$ at a given point $a\in X$, i.e.\ we want to compute
$h^0\big(\cO(m[a])\big)$. As we shall see soon, these numbers may
depend on $a$ only if $m$ is small. We have
$c_1\big(K(-m[a])\big)=2g-2-m$, so the degree is $<0$ and
$h^0\big(K(-m[a])\big)=0$ for $m\ge 2g-1$ by~10.7~a).
Cor.~10.12 implies
$$h^0\big(\cO(m[a])\big)=m-g+1~~~{\rm for}~~m\ge 2g-1.\leqno(10.18)$$
It remains to compute $h^0\big(K(-m[a])\big)$ for $0\le m\le 2g-2$ and
\hbox{$g\ge 1$.} Let $u_1,\ldots,u_g$ be a basis of $H^0(X,K)$ and let $z$
be a complex coordinate centered at~$a$. Any germ $u\in\cO(K)_a$
can be written \hbox{$u=U(z)\,dz$} with \hbox{$U(z)=\sum_{m\in\bN}{1\over m!}
U^{(m)}(a)z^m\,dz$.} The unique non zero stalk of the quotient sheaf
\hbox{$\cO\big(K(-m[a])\big)/\cO\big(K(-(m+1)[a])\big)$}
is canonically isomorphic to $K^{m+1}_a$ via the map
$u\mapsto U^{(m)}(a)(dz)^{m+1}$, which is independant of the choice of~$z$.
Hence \hbox{$\bigwedge^g\big(\cO(K)/\cO(K-g[a])\big)\simeq K_a^{1+2+\ldots+g}$}
and the {\it Wronskian}
$$W(u_1,\ldots,u_g)=\left|\matrix{
~U_1(z)&\ldots&U_g(z)\cr
~U'_1(z)&\ldots&U'_g(z)\cr
\vdots&&\vdots\cr
~U_1^{(g-1)}(z)\vphantom{\displaystyle\sum^a}&\ldots&U_g^{(g-1)}(z)~\cr}
\right|\,dz^{1+2+\ldots+g}\leqno(10.19)$$
defines a global section 
$W(u_1,\ldots,u_g)\in H^0(X,K^{g(g+1)/2})$. At the given point $a$, 
we can find linear combinations $\wt u_1,\ldots,\wt u_g$ of $u_1,\ldots,u_g$ such that
$$\wt u_j(z)=\big(z^{s_j-1}+{\rm O}(z^{s_j})\big)dz,~~~~s_1<\ldots<s_g.$$
We know that not all sections of $K$ vanish at $a$ and that $c_1(K)=2g-2$,
thus  $s_1=1$ and $s_g\le 2g-1$. We have
$W(\wt u_1,\ldots,\wt u_g)\sim W(z^{s_1-1}dz,\ldots,z^{s_g-1}dz)$ at point $a$, and 
an easy induction on $\sum s_j$ combined with differentiation in $z$ yields
$$W(z^{s_1-1}dz,\ldots,z^{s_g-1}dz)=C\,z^{s_1+\ldots+s_g-g(g+1)/2}\,
dz^{g(g+1)/2}$$
for some positive integer constant $C$. In particular, $W(u_1,\ldots,u_g)$ is
not identically zero and vanishes at $a$ with multiplicity
$$\mu_a=s_1+\ldots+s_g-g(g+1)/2>0\leqno(10.20)$$
unless $s_1=1$, $s_2=2$, $\ldots$, $s_g=g$. Now, we have
$$h^0\big(K(-m[a])\big)={\rm card}\{j\,;\,s_j>m\}
=g-{\rm card}\{j\,;\,s_j\le m\}$$
and Cor.~10.12 gives
$$h^0\big(\cO(m[a])\big)=m+1-{\rm card}\{j\,;\,s_j\le m\}.\leqno(10.21)$$
If $a$ is not a zero of $W(u_1,\ldots,u_g)$, we find
$$\cases{
h^0\big(\cO(m[a])\big)=1&for~~$m\le g$,\cr
h^0\big(\cO(m[a])\big)=m+1-g&for~~$m>g$.\cr}\leqno(10.22)$$
The zeroes of $W(u_1,\ldots,u_g)$ are called the {\it Weierstrass points} of $X$,
and the associated {\it Weierstrass sequence} is the sequence 
$w_m=h^0\big(\cO(m[a])\big)$, $m\in\bN$. We have 
$w_{m-1}\le w_m\le w_{m-1}+1$ and $s_1<\ldots<s_g$ are precisely the integers
$m\ge 1$ such that $w_m=w_{m-1}$. The numbers $s_j\in\{1,2,\ldots,2g-1\}$ are 
called the {\it gaps} and $\mu_a$ the {\it weight} of the Weierstrass point 
$a$. Since $W(u_1,\ldots,u_g)$ is a section of $K^{g(g+1)/2}$, 
Hurwitz' formula implies
$$\sum_{a\in X}\mu_a=c_1(K^{g(g+1)/2})=g(g+1)(g-1).\leqno(10.23)$$
In particular, a curve of genus $g$ has at most $g(g+1)(g-1)$ Weierstrass 
points.

\section{11}{Hodge-Fr\"olicher Spectral Sequence}
Let $X$ be a {\it compact} complex $n$-dimensional manifold. We consider the
double complex $K^{p,q}=\cC^\infty(X,\Lambda^{p,q}T^\star_X)$, $d=d'+d''$. The 
Hodge-Fr\"olicher spectral sequence is by definition the spectral
sequence associated to this double complex (cf.\ IV-11.9). It starts with
$$E^{p,q}_1=H^{p,q}(X,\bC)\leqno(11.1)$$
and the limit term $E^{p,q}_\infty$ is the graded module associated to a 
filtration of the De Rham cohomology group $H^k(X,\bC)$, $k=p+q$.
In particular, if the numbers $b_k$ and $h^{p,q}$ are still defined
as in (8.11), we have
$$b_k=\sum_{p+q=k}\dim E^{p,q}_\infty\le\sum_{p+q=k}\dim E^{p,q}_1=
\sum_{p+q=k}h^{p,q}.\leqno(11.2)$$
The equality is equivalent to the degeneration of the spectral sequence at 
$E^\bu_1$. As a consequence, the Hodge-Fr\"olicher spectral
sequence of a compact K\"ahler manifold degenerates in $E^\bu_1$.

\claim{(11.3) Theorem and Definition} The existence of an isomorphism
$$H^k_\DR(X,\bC)\simeq\bigoplus_{p+q=k}H^{p,q}(X,\bC)$$
is equivalent to the degeneration of the Hodge-Fr\"olicher spectral
sequence at~$E_1$. In this case, the isomorphism is canonically defined
and we say that $X$ admits a Hodge decomposition.\qed
\endclaim

In general, interesting informations can be deduced from the spectral
sequence. Theorem~IV-11.8 shows in particular that 
$$b_1\ge\dim E^{1,0}_2+(\dim E^{0,1}_2-\dim E^{2,0}_2)_+.\leqno(11.4)$$
However, $E^{1,0}_2$ is the central cohomology group in the sequence
$$d_1=d':E^{0,0}_1\longrightarrow E^{1,0}_1\longrightarrow E^{2,0}_1,$$
and as $E^{0,0}_1$ is the space of holomorphic functions on $X$, the
first map $d_1$ is zero (by the maximum principle, holomorphic functions
are constant on each connected component of $X\,$). Hence
$\dim E^{1,0}_2\ge h^{1,0}-h^{2,0}$. Similarly, $E^{0,1}_2$ is
the kernel of a map $E^{0,1}_1\to E^{1,1}_1$, thus
$\dim E^{0,1}_2\ge h^{0,1}-h^{1,1}$. By (11.4) we obtain
$$b_1\ge (h^{1,0}-h^{2,0})_++(h^{0,1}-h^{1,1}-h^{2,0})_+.\leqno(11.5)$$
Another interesting relation concerns the topological Euler-Poincar\'e
characteristic
$$\chi_{{\rm top}}(X)=b_0-b_1+\ldots-b_{2n-1}+b_{2n}.$$
We need the following simple lemma.

\claim{(11.6) Lemma} Let $(C^\bu,d)$ a bounded complex of finite dimensional
vector spaces over some field. Then, the Euler characteristic 
$$\chi(C^\bu)=\sum(-1)^q\dim C^q$$
is equal to the Euler characteristic $\chi\big(H^\bu(C^\bu)\big)$ of the
cohomology module. 
\endclaim

\proof{Proof} Set
$$c_q=\dim C^q,\qquad z_q=\dim Z^q(C^\bu),\qquad b_q=\dim B^q(C^\bu),\qquad
h_q=\dim H^q(C^\bu).$$
Then
$$c_q=z_q+b_{q+1},~~~~h_q=z_q-b_q.$$
Therefore we find
$$\sum(-1)^q\,c_q=\sum(-1)^q\,z_q-\sum(-1)^q\,b_q=\sum(-1)^q\,h_q.
\eqno{\square}$$
\endproof

In particular, if the term $E^\bu_r$ of the spectral sequence of a
filtered complex $K^\bu$ is a bounded and finite dimensional complex, we 
have
$$\chi(E^\bu_r)=\chi(E^\bu_{r+1})=\ldots=\chi(E^\bu_\infty)=
\chi\big(H^\bu(K^\bu)\big)$$
because $E^\bu_{r+1}=H^\bu(E^\bu_r)$ and $\dim E^l_\infty=\dim H^l(K^\bu)$.
In the Hodge-Fr\"olicher spectral sequence, we have
$\dim E^l_1=\sum_{p+q=l}h^{p,q}$, hence:

\claim{(11.7) Theorem} For any compact complex manifold $X$, one has
$$\chi_{{\rm top}}(X)=\sum_{0\le k\le 2n}(-1)^kb_k=\sum_{0\le p,q\le n}
(-1)^{p+q}h^{p,q}.$$
\endclaim

\section{12}{Effect of a Modification on Hodge Decomposition}
In this section, we show that the existence of a Hodge decomposition on
a compact complex manifold $X$ is guaranteed as soon as there exists
such a decomposition on a modification $\wt X$ of~$X$
(see II-??.?? for the Definition). This leads us to extend Hodge theory
to a class of manifolds which are non necessarily K\"ahler, the so called
Fujiki class $(\cC)$ of manifolds bimeromorphic to K\"ahler
manifolds.

\subsection{12.1}{Sheaf Cohomology Reinterpretation of
$H_\BC^{p,q}(X,\bC)$}
We first give a description of $H_\BC^{p,q}(X,\bC)$ in terms of
the hypercohomology of a suitable complex of sheaves. This
interpretation, combined with the analogue of the Hodge-Fr\"olicher
spectral sequence, will imply in particular that $H_\BC^{p,q}(X,\bC)$
is always finite dimensional when $X$ is compact.
Let us denote by $\cE^{p,q}$ the sheaf of germs of $\cC^\infty$ forms of
bidegree $(p,q)$, and by $\Omega^p$ the sheaf of germs of
holomorphic $p$-forms on $X$. 
For a fixed bidegree $(p_0,q_0)$, we let $k_0=p_0+q_0$ and we
introduce a complex of sheaves $(\cL^\bu_{p_0,q_0},\delta)$, also 
denoted $\cL^\bu$ for simplicity, such that
$$\eqalign{
\cL^k&=\bigoplus_{p+q=k,p<p_0,q<q_0}\cE^{p,q}~~~{\rm for}~~
k\le k_0-2,\cr
\cL^{k-1}&=\bigoplus_{p+q=k,p\ge p_0,q\ge q_0}\cE^{p,q}~~~{\rm for}~~
k\ge k_0.\cr}$$
The differential $\delta^k$ on $\cL^k$ is chosen equal
to the exterior derivative $d$ for $k\ne k_0-2$ (in the case
$k\le k_0-3$, we neglect the components which fall outside $\cL^{k+1}$), 
and we set
$$\delta^{k_0-2}=d'd'':\cL^{k_0-2}=\cE^{p_0-1,q_0-1}\longrightarrow\cL^{k_0-1}=
\cE^{p_0,q_0}.$$
We find in particular $H_\BC^{p_0,q_0}(X,\bC)=H^{k_0-1}\big(\cL^\bu(X)\big)$.
We observe that $\cL^\bu$ has subcomplexes $(\cS^{\prime\,\bu},d')$ and
$(\cS^{\prime\prime\,\bu},d'')$ defined by
$$\cmalign{
\cS^{\prime\, k}&=\Omega_X^k~~~{\rm for}~~0\le k\le p_0-1,~~~~
\cS^{\prime\, k}&=0~~~{\rm otherwise},\cr
\cS^{\prime\prime\, k}&=\ol{\Omega_X^k}~~~{\rm for}~~0\le k\le q_0-1,~~~~
\cS^{\prime\prime\, k}&=0~~~{\rm otherwise}.\cr}$$
If $p_0=0$ or $q_0=0$ we set instead $\cS^{\prime\,0}=\bC$ or
$\cS^{\prime\prime\,0}=\bC$, and take the other components to be zero.
Finally, we let $\cS^\bu=\cS^{\prime\,\bu}+\cS^{\prime\prime\,\bu}
\subset\cL^\bu$ (the sum is direct except for $\cS^0$); we denote by 
$\cM^\bu$ the sheaf complex defined in the same way as $\cL^\bu$,
except that the sheaves $\cE^{p,q}$ are replaced by the sheaves of currents 
$\cD'_{n-p,n-q}$.

\claim{(12.1) Lemma} The inclusions $\cS^\bu\subset\cL^\bu\subset\cM^\bu$
induce isomorphisms 
$$\cH^k(\cS^\bu)\simeq\cH^k(\cL^\bu)\simeq\cH^k(\cM^\bu),$$
and these cohomology sheaves vanish for $k\ne 0,p_0-1,q_0-1$.
\endclaim

\proof{Proof} We will prove the result only for the inclusion $\cS^\bu\subset
\cL^\bu$,
the other case $\cS^\bu\subset\cM^\bu$ is identical. Let us denote by
$\cZ^{p,q}$ the sheaf of $d''$-closed differential forms of bidegree
$(p,q)$. We consider the filtration 
$$F_p(\cL^k)=\cL^k\cap\bigoplus_{r\ge p}\cE^{r,\bu}$$ and the induced 
filtration on $\cS^\bu$. In the case of $\cL^\bu$, the first spectral 
sequence has the following terms $E^\bu_0$ and $E^\bu_1\,$:
$$\cmalign{
&{\rm if}~~p<p_0~~~~&E^{p,\bu}_0~:~~~0\longrightarrow\cE^{p,0}\buildo d''\over\longrightarrow
\cE^{p,1}\longrightarrow\cdots\buildo d''\over\longrightarrow\cE^{p,q_0-1}\longrightarrow 0,\cr
&{\rm if}~~p\ge p_0~~~~&E^{p,\bu}_0~:~~~0\longrightarrow\cE^{p,q_0}\buildo d''\over\longrightarrow
\cE^{p,q_0+1}\longrightarrow\cdots\longrightarrow\cE^{p,q}\buildo d''\over\longrightarrow\cdots,\cr
&{\rm if}~~p<p_0~~~~&E^{p,0}_1=\Omega_X^p,~~E^{p,q_0-1}_1\simeq\cZ^{p,q_0},~~
E^{p,q}_1=0~~~{\rm for}~~q\ne 0,q_0-1,\cr
&{\rm if}~~p\ge p_0~~~~&E^{p,q_0-1}_1=\cZ^{p,q_0},~~
E^{p,q}_1=0~~~{\rm for}~~q\ne q_0-1.\cr}$$
The isomorphism in the third line is given by
$$\cE^{p,q_0-1}/d''\cE^{p,q_0-2}\simeq d''\cE^{p,q_0-1}\simeq\cZ^{p,q_0}.$$
The map $d_1:E^{p_0-1,q_0-1}_1\longrightarrow E^{p_0,q_0-1}_1$ is induced by
$d'd''$ acting on $\cE^{p_0-1,q_0-1}$, but thanks to the previous
identification, this map becomes $d'$ acting on $\cZ^{p_0-1,q_0}$.
Hence $E^\bu_1$ consists of two sequences
$$\eqalign{
&E_1^{\bu,0}~:~~0\longrightarrow\Omega_X^0\buildo d'\over\longrightarrow\Omega_X^1\longrightarrow\cdots\buildo d'\over
\longrightarrow\Omega_X^{p_0-1}\longrightarrow 0,\cr 
&E_1^{\bu,q_0-1}~:~~0\longrightarrow\cZ^{0,q_0}\buildo d'\over\longrightarrow\cZ^{1,q_0}\longrightarrow\cdots\longrightarrow
\cZ^{p,q_0}\buildo d'\over\longrightarrow\cdots~;\cr}$$
if these sequences overlap ($q_0=1$), only the second one has to be 
considered. The term $E^\bu_1$ in the spectral sequence of $\cS^\bu$ has the 
same first line, but the second is reduced to $E^{0,q_0-1}_1
=\ol{d\Omega_X^{q_0-2}}$ (resp.\ $=\bC$ for $q_0=1$). Thanks to
Lemma~12.2 below, we see that the two spectral sequences coincide in
$E^\bu_2$, with at most three non zero terms:
$$E^{0,0}_2=\bC,~~~~E^{p_0-1,0}_2=d\Omega_X^{p_0-2}~~{\rm for}~~p_0\ge 2,~~~~
E^{0,q_0-1}_2=\ol{d\Omega_X^{q_0-2}}~~{\rm for}~~q_0\ge 2.$$
Hence $\cH^k(\cS^\bu)\simeq\cH^k(\cL^\bu)$ and these sheaves vanish
for $k\ne 0,p_0-1,q_0-1$.\qed
\endproof

\claim{(12.2) Lemma} The complex of sheaves
$$0\longrightarrow\cZ^{0,q_0}\buildo d'\over\longrightarrow\cZ^{1,q_0}\longrightarrow\cdots\longrightarrow\cZ^{p,q_0}
\buildo d'\over\longrightarrow\cdots$$
is a resolution of~ $\ol{d\Omega_X^{q_0-1}}$ for $q_0\ge 1$, resp.\ of~ $\bC$
for $q_0=0$.
\endclaim

\proof{Proof} Embed $\cZ^{\bu,q_0}$ in the double complex
$$K^{p,q}=\cE^{p,q}~~~{\rm for}~~q<q_0,~~~~
K^{p,q}=0~~~{\rm for}~~q\ge q_0.$$
For the first fitration of $K^\bu$, we find
$$E^{p,q_0-1}_1=\cZ^{p,q_0},~~~~E^{p,q}_1=0~~~{\rm for}~~q\ne q_0-1~$$
The second fitration gives $\wt E^{p,q}_1=0$ for $q\ge 1$ and
$$\wt E^{p,0}_1=H^0(K^{\bu,p})=
\cases{\ol{H^0(\cE^{p,\bu})}=\ol{\Omega_X^p}&for $p\le q_0-1$\cr
       0&for $p\ge q_0$,\cr}$$
thus the cohomology of $\cZ^{\bu,q_0}$ coincides with that of
$(\ol{\Omega_X^p},d)_{0\le p<q_0}$.\qed
\endproof

Lemma IV-11.10 and formula (IV-12.9) imply
$$\leqalignno{
{\Bbb H}^k(X,\cS^\bu)&{}\simeq{\Bbb H}^k(X,\cL^\bu)\simeq
{\Bbb H}^k(X,\cM^\bu)&(12.3)\cr
&{}\simeq H^k\big(\cL^\bu(X)\big)\simeq H^k\big(\cM^\bu(X)\big)\cr}$$
because the sheaves $\cL^k$ and $\cM^k$ are soft. In particular,
the group $H_\BC^{p,q}(X,\bC)$ can be computed either by means
of $\cC^\infty$ differential forms or by means of currents. This property
also holds for the De Rham or Dolbeault groups $H^k(X,\bC)$,
$H^{p,q}(X,\bC)$, as was already remarked in \S IV-6. Another important 
consequence of (12.3) is:

\claim{(12.4) Theorem} If $X$ is compact, then $\dim H_\BC^{p,q}(X,\bC)<+\infty$.
\endclaim

\proof{Proof} We show more generally that the hypercohomology groups
${\Bbb H}^k(X,\cS^\bu)$ are finite dimensional. As there is an exact
sequence
$$0\longrightarrow\bC\longrightarrow\cS^{\prime\,\bu}\oplus\cS^{\prime\prime\,\bu}\longrightarrow
\cS^\bu\longrightarrow 0$$
and a corresponding long exact sequence for hypercohomology groups,
it is enough to show that the groups
${\Bbb H}^k(X,\cS^{\prime\,\bu})$ are finite
dimensional. This property is proved for $\cS^{\prime\,\bu}=
\cS^{\prime\,\bu}_{p_0}$ by induction on $p_0$. For $p_0=0$ or $1$, the
complex $\cS^{\prime\,\bu}$ is reduced to its term $\cS^{\prime\,0}$, thus
$${\Bbb H}^k(X,\cS^\bu)=H^k(X,\cS^{\prime\,0})=\cases{
H^k(X,\bC)&for $p_0=0$\cr
H^k(X,\cO)&for $p_0=1$\cr}$$
and this groups are finite dimensional. In general, we have an exact
sequence
$$0\longrightarrow\Omega_X^{p_0}\longrightarrow\cS^\bu_{p_0+1}\longrightarrow\cS^\bu_{p_0}\longrightarrow 0$$
where $\Omega_X^{p_0}$ denotes the subcomplex of $\cS^\bu_{p_0+1}$ reduced to 
one term in degree~$p_0$. As
$${\Bbb H}^k(X,\Omega_X^{p_0})=H^{k-p_0}(X,\Omega_X^{p_0})=
H^{p_0,k-p_0}(X,\bC)$$
is finite dimensional, the Theorem follows.\qed
\endproof

\claim{(12.5) Definition} We say that a compact manifold admits a
strong Hodge decomposition if the natural maps
$$H_\BC^{p,q}(X,\bC)\longrightarrow H^{p,q}(X,\bC),~~~~
\bigoplus_{p+q=k}H_\BC^{p,q}(X,\bC)\longrightarrow H^k(X,\bC)$$
are isomorphisms.
\endclaim

This implies of course that there are natural isomorphisms
$$H^k(X,\bC)\simeq\bigoplus_{p+q=k}H^{p,q}(X,\bC),~~~~H^{q,p}(X,\bC)\simeq
\ol{H^{p,q}(X,\bC)}$$
and that the Hodge-Fr\"olicher spectral sequence degenerates in $E^\bu_1$.
It follows from \S$\,$8 that all K\"ahler manifolds admit a strong Hodge
decomposition.

\subsection{12.2}{Direct and Inverse Image Morphisms}
Let $F:X\longrightarrow Y$ be a holomorphic map between complex analytic manifolds of
respective dimensions $n,m$, and $r=n-m$. We have pull-back morphisms
$$\cmalign{
&F^\star~:~~&\hfill H^k(Y,\bC)&\longrightarrow H^k(X,\bC),\cr
&F^\star~:~~&\hfill H^{p,q}(Y,\bC)&\longrightarrow H^{p,q}(X,\bC),\cr
&F^\star~:~~&\hfill H_\BC^{p,q}(Y,\bC)&\longrightarrow H_\BC^{p,q}(X,\bC),\cr}
\leqno(12.6)$$
commuting with the natural morphisms (8.2), (8.3).

Assume now that $F$ is {\it proper}. Theorem~I-1.14 shows that one
can define direct image morphisms
$$F_\star~:~~\cD'_k(X)\longrightarrow\cD'_k(Y),~~~~
  F_\star~:~~\cD'_{p,q}(X)\longrightarrow\cD'_{p,q}(Y),$$
commuting with $d',d''$. To $F_\star$ therefore correspond cohomology 
morphisms
$$\cmalign{
&F_\star~:~~&\hfill H^k(X,\bC)&\longrightarrow H^{k-2r}(Y,\bC),\cr
&F_\star~:~~&\hfill H^{p,q}(X,\bC)&\longrightarrow H^{p-r,q-r}(Y,\bC),\cr
&F_\star~:~~&\hfill H_\BC^{p,q}(X,\bC)&\longrightarrow H_\BC^{p-r,q-r}(Y,\bC),\cr}
\leqno(12.7)$$
which commute also with (8.2), (8.3). In addition, I-1.14~c)
implies the {\it adjunction formula}
$$F_\star(\alpha\smallsmile F^\star\beta)=(F_\star\alpha)\smallsmile\beta
\leqno(12.8)$$
whenever $\alpha$ is a cohomology class (of any of the three above types) 
on $X$, and $\beta$ a cohomology class (of the same type) on $Y$.

\subsection{12.3}{Modifications and the Fujiki Class ($\cC$)}
Recall that a modification of a compact manifold $X$ is a
holomorphic map $\mu:\wt X\longrightarrow X$ such that
\smallskip
\item{\rm i)} $\wt X$ is a compact complex manifold of the
same dimension as $X\,$;
\smallskip
\item{\rm ii)} there exists an analytic subset $S\subset X$ of 
codimension $\ge 1$ such that $\mu:\wt X\setminus
\mu^{-1}(S)\longrightarrow X\setminus S$ is a biholomorphism.
\vskip0pt

\claim{(12.9) Theorem} If $\wt X$ admits a strong Hodge decomposition, 
and if $\mu:\wt X\longrightarrow X$ is a modification, then $X$ also admits a
strong Hodge decomposition.
\endclaim

\proof{Proof} We first observe that $\mu_\star\mu^\star f=f$ for every smooth form
$f$ on $Y$. In fact, this property is equivalent to the equality
$$\int_Y (\mu_\star\mu^\star f)\wedge g=\int_X\mu^\star(f\wedge g)=\int_Y
f\wedge g$$
for every smooth form $g$ on $Y$, and this equality is clear because 
$\mu$ is a biholomorphism outside sets of Lebesgue measure $0$. Consequently,
the induced cohomology morphism
$\mu_\star$ is surjective and $\mu^\star$ is injective (but these maps need 
not be isomorphisms).  Now, we have commutative diagrams
$$\cmalign{
&H_\BC^{p,q}(\wt X,\bC)\longrightarrow&H^{p,q}(\wt X,\bC),~~~~
&\smash{\displaystyle\bigoplus_{p+q=k}}
H_\BC^{p,q}(\wt X,\bC)\longrightarrow&H^k(\wt X,\bC)
\phantom{\raise-6pt\hbox{$\Big)$}}\cr
&~~~\mu_\star\big\downarrow\big\uparrow\mu^\star\hfill
&~~~\mu_\star\big\downarrow\big\uparrow\mu^\star\hfill
&\qquad~~\mu_\star\big\downarrow\big\uparrow\mu^\star\hfill
&~~\mu_\star\big\downarrow\big\uparrow\mu^\star\hfill\cr
\noalign{\vskip6pt}
&H_\BC^{p,q}(X,\bC)\longrightarrow&H^{p,q}(X,\bC),~~~~
&\displaystyle\bigoplus_{p+q=k}H_\BC^{p,q}(X,\bC)\longrightarrow&H^k(X,\bC)
\phantom{\Big)}\cr}$$
with either upward or downward vertical arrows. Hence the surjectivity or
injectivity of the top horizontal arrows implies that
of the bottom horizontal arrows.\qed
\endproof

\claim{(12.10) Definition} A manifold $X$ is said to be in the
Fujiki class $(\cC)$ if $X$ admits a K\"ahler modification $\wt X$.
\endclaim

By Th.~12.9, Hodge decomposition still holds for a manifold in the
class $(\cC)$. We will see later that there exist non-K\"ahler manifolds 
in $(\cC)$, for example all non projective Moi$\check{\rm s}$ezon 
manifolds (cf.\ \S ?.?). The class $(\cC)$ has been first introduced
in \cite{Fujiki~1978}.


\chapter{Chapter VII}{Positive Vector Bundles and Vanishing Theorems}
\begineightpoint
In this chapter, we prove a few vanishing theorems for hermitian vector
bundles over {\it compact} complex manifolds. All these theorems are
based on an a priori inequality for \hbox{$(p,q)$-forms} with values in
a vector bundle, known as the Bochner-Kodaira-Nakano inequality. This
inequality naturally leads to several positivity notions for the
curvature of a vector bundle (\cite{Kodaira 1953, 1954}, 
\cite{Griffiths 1969} and \cite{Nakano 1955, 1973}). The corresponding 
algebraic notion of ampleness
introduced by Grothendieck \cite{EGA 1960-67} and developped further
by \cite{Nakai 1963} and \cite{Hartshorne 1966} is also
discussed. The differential geometric techniques yield optimal
vanishing results in the case of line bundles (Kodaira-Akizuki-Nakano
and Girbau vanishing theorems) and also some partial results in the
case of vector bundles  (Nakano vanishing theorem). As an illustration,
we compute the cohomology  groups $H^{p,q}(\bP^n,\cO(k))\,$; much
finer results will be obtained in chapters 8--11. Finally, the Kodaira
vanishing theorem is combined with a blowing-up technique in order to
establish the projective embedding theorem for manifolds admitting
a Hodge metric.
\endeightpoint

\section{1}{Bochner-Kodaira-Nakano Identity}
Let $(X,\omega)$ be a hermitian manifold, $\dim_\bC X=n$, and let $E$
be a hermitian holomorphic vector bundle of rank $r$ over $X$. We denote by
$D=D'+D''$ its Chern connection (or $D_E$ if we want to specify the
bundle), and by $\delta=\delta'+\delta''$ the formal adjoint operator
of $D$. The operators $L,\Lambda$ of chapter 6 are extended to vector
valued forms in $\Lambda^{p,q}T^\star X\otimes E$ by taking their
tensor product with  $\Id_E$. The following result extends
the commutation relations of chapter 6 to the 
case of bundle valued operators.

\claim{(1.1) Theorem} If $\tau$ is the operator of type $(1,0)$ defined by 
$\tau = [\Lambda, d'\omega]$ on $\cC^\infty_{\bu,\bu}(X,E)$, then
\medskip\noindent
$\cmalign{
\hbox{\rm a)}\hfill\qquad &[\delta''_E,L] &= &\ii(D'_E+\tau),\cr
\hbox{\rm b)}\hfill\qquad &[\delta'_E,L] &= -&\ii(D''_E+\ol \tau),\cr
\hbox{\rm c)}\hfill\qquad &[\Lambda,D''_E] &= -&\ii(\delta'_E+\tau^\star),\cr
\hbox{\rm d)}\hfill\qquad &[\Lambda,D'_E] &= &\ii(\delta''_E+\ol \tau^\star).
\cr}$
\endclaim

\proof{Proof} Fix a point $x_0$ in X and a coordinate system $z=(z_1,\ldots,z_n)$
centered at~$x_0$. Then Prop.~V-12.?? shows the existence of a
normal coordinate frame $(e_\lambda)$ at~$x_0$.
Given any section $s=\sum_\lambda\sigma_\lambda\otimes e_\lambda\in
\cC^\infty_{p,q}(X,E)$, it is easy to check that the operators $D_E$,
$\delta_E'',\,\ldots~$ have Taylor expansions of the type
$$D_Es=\sum_\lambda d\sigma_\lambda\otimes e_\lambda+O(|z|),~~~~
\delta_E''s=\sum_\lambda \delta''\sigma_\lambda\otimes e_\lambda+O(|z|),
~\ldots$$
in terms of the scalar valued operators $d$, $\delta$, $\ldots$.
Here the terms $O(|z|)$ depend on the curvature coefficients of $E$.
The proof of Th.~1.1 is then reduced to the case of scalar valued
operators, which is granted by Th.~VI-10.1.\qed
\endproof

The Bochner-Kodaira-Nakano identity expresses the antiholomorphic
La\-pla\-ce operator $\Delta''=D''\delta''+\delta''D''$ acting on
$\cC^\infty_{\bu,\bu}(X,E)$ in terms of its conjugate operator
$\Delta'=D'\delta'+\delta'D'$, plus some extra terms involving
the curvature of $E$ and the torsion of the metric $\omega$ (in case
$\omega$ is not K\"ahler). Such identities appear frequently in
riemannian geometry (Weitzenb\"ock formula).

\claim{(1.2) Theorem} $\Delta''=\Delta'+[\ii\Theta(E),\Lambda]+[D',\tau^\star]-
[D'',\ol\tau^\star]$.
\endclaim

\proof{Proof} Equality 1.1~d) yields $\delta''=-\ii[\Lambda,D']-
\ol\tau^\star$, hence
$$\Delta''=[D'',\delta'']=-\ii[D'',\big[\Lambda,D']\big]-[D'',\ol\tau^\star].$$
The Jacobi identity~VI-10.2 and relation 1.1~c) imply
$$\big[D'',[\Lambda,D']\big]=\big[\Lambda,[D',D'']]+\big[D',[D'',\Lambda]\big]
=[\Lambda,\Theta(E)]+\ii[D',\delta'+\tau^\star],$$
taking into account that $[D',D'']=D^2=\Theta(E)$. Theorem~1.2 follows.\qed
\endproof

\claim{(1.3) Corollary (\cite{Akizuki-Nakano 1955})} If $\omega$ is
K\"ahler, then
$$\Delta''=\Delta'+[\ii\Theta(E),\Lambda].$$
\endclaim

In the latter case, $\Delta''-\Delta'$ is therefore an operator of order $0$
closely related to the curvature of $E$. When $\omega$ is not K\"ahler,
Formula~1.2 is not really satisfactory, because it involves the first order
operators $[D',\tau^\star]$ and $[D'',\ol\tau^\star]$. In fact,
these operators can be combined with $\Delta'$ in order to yield a new
positive self-adjoint operator $\Delta'_\tau$.

\claim{(1.4) Theorem (\cite{Demailly 1985})} The operator
$\Delta'_\tau=[D'+\tau,\delta'+ \tau^\star]$ is a positive and formally
self-adjoint operator with the same principal part as the Laplace
operator~$\Delta'$. Moreover
$$\Delta''=\Delta'_\tau+[\ii\Theta(E),\Lambda]+T_\omega,$$
where $T_\omega$ is an operator of order $0$ depending only on the
torsion of the hermitian metric $\omega\,:$
$$T_\omega=\Big[\Lambda,\big[\Lambda,{\ii\over 2}d'd''\omega\big]\Big]-
\big[d'\omega,(d'\omega)^\star\big].$$
\endclaim

\proof{Proof} The first assertion is clear, because the equality
$(D'+\tau)^\star=\delta'+\tau^\star$ implies the self-adjointness of 
$\Delta'_\tau$ and
$$\Ll\Delta'_\tau u,u\Gg=\|D'u+\tau u\|^2+\|\delta'u+\tau^\star u\|^2\ge 0$$
for any compactly supported form $u\in \cC^\infty_{p,q}(X,E)$. In order to prove
the formula, we need two lemmas.
\endproof

\claim{(1.5) Lemma} {\rm a)}$\qquad[L,\tau]=3d'\omega,$ $\qquad~~$
{\rm b)}$\qquad[\Lambda,\tau]=-2\ii\ol\tau^\star.$
\endclaim

\proof{Proof} a) Since $[L,d'\omega]=0$, the Jacobi identity implies
$$[L,\tau]=\big[L,[\Lambda,d'\omega]\big]=-\big[d'\omega,[L,\Lambda]\big]=
3d'\omega,$$
taking into account Cor.~VI-10.4 and the fact that $d'\omega$ is of
degree $3$.
\medskip
\noindent b) By 1.1~a) we have $\tau=-\ii[\delta'',L]-D'$, hence
$$[\Lambda,\tau]=-i\big[\Lambda,[\delta'',L]\big]-[\Lambda,D']=
-i\big(\big[\Lambda,[\delta'',L]\big]+\delta''+\ol\tau^\star\big).$$
Using again VI-10.4 and the Jacobi identity, we get
$$\eqalign{
\big[\Lambda,[\delta'',L]\big]&=-\big[L,[\Lambda,\delta'']\big]-\big[\delta'',
[L,\Lambda]\big]\cr
&=-\big[[d'',L],\Lambda\big]^\star-\delta''=-[d''\omega,\Lambda]^\star-\delta''
=\ol\tau^\star-\delta''.\cr}$$
A substitution in the previous equality gives $[\Lambda,\tau]=
-2\ii\ol\tau^\star$.\qed
\endproof

\claim{(1.6) Lemma} The following identities hold:
\smallskip
\item{\rm a)} $[D',\ol\tau^\star]=-[D',\delta'']=[\tau,\delta''],$
\smallskip
\item{\rm b)} $-[D'',\ol\tau^\star]=[\tau,\delta'+\tau^\star]+T_\omega.$
\endclaim

\proof{Proof} a) The Jacobi identity implies
$$-\big[D',[\Lambda,D']\big]+\big[D',[D',\Lambda]\big]+\big[\Lambda,[D',D']
\big]=0,$$
hence $-2\big[D',[\Lambda,D']\big]=0$ and likewise $\big[\delta'',[\delta'',L]
\big]=0$. Assertion a) is now a consequence of 1.1~a) and d).
\medskip
\noindent b) In order to verify b), we start from the equality
$\ol\tau^\star={\ii\over 2}[\Lambda,\tau]$ provided by Lemma~1.5 b). 
It follows that
$$[D'',\ol\tau^\star]={\ii\over 2}\big[D'',[\Lambda,\tau]\big].\leqno(1.7)$$
The Jacobi identity will now be used several times. One obtains
$$\leqalignno{
\qquad\big[D'',[\Lambda&,\tau]\big]=\big[\Lambda,[\tau,D'']\big]+
\big[\tau,[D'',\Lambda]\big]~;&(1.8)\cr
[\tau,D'']&=[D'',\tau]=\big[D'',[\Lambda,d'\omega]\big]
=\big[\Lambda,[d'\omega,D'']\big]+\big[d'\omega,[D'',\Lambda]\big]&(1.9)\cr
&=[\Lambda,d''d'\omega]+[d'\omega,A]\cr}$$
with $A=[D'',\Lambda]=\ii(\delta'+\tau^\star)$. From (1.9) we deduce
$$\big[\Lambda,[\tau,D'']\big]=\big[\Lambda,[\Lambda,d''d'\omega]\big]+
\big[\Lambda,[d'\omega,A]\big].\leqno(1.10)$$
Let us compute now the second Lie bracket in the right hand side of (1.10:
$$\leqalignno{
\qquad\big[\Lambda,[d'\omega,A]\big]&=\big[A,[\Lambda,d'\omega]\big]-
\big[d'\omega,[A,\Lambda]\big]=[\tau,A]+\big[d'\omega,[\Lambda,A]\big]~;
&(1.11)\cr
[\Lambda,A]&=\ii[\Lambda,\delta'+\tau^\star]=\ii[D'+\tau,L]^\star.&(1.12)\cr}$$
Lemma 1.5 b) provides $[\tau,L]=-3d'\omega$, and it is clear that
$[D',L]=d'\omega$. Equalities (1.12) and (1.11) yield therefore
$$\leqalignno{
[\Lambda,A]&=-2\ii(d'\omega)^\star,\cr
\big[\Lambda,[d'\omega,A]\big]&=\big[\tau,[D'',\Lambda]\big]-2\ii
[d'\omega,(d'\omega)^\star].&(1.13)\cr}$$
Substituting (1.10) and (1.13) in (1.8) we get
$$\leqalignno{
\big[D'',[\Lambda,\tau]\big]&=\big[\Lambda,[\Lambda,d''d'\omega]\big]+
2\big[\tau,[D'',\Lambda]\big]-2\ii\big[d'\omega,(d'\omega)^\star\big]&(1.14)\cr
&=2\ii\big(T_\omega+[\tau,\delta'+\tau^\star]\big).\cr}$$
Formula b) is a consequence of (1.7) and (1.14).\qed
\endproof

Theorem 1.4 follows now from Th.~1.2 if Formula~1.6 b) is
rewritten 
$$\Delta'+[D',\tau^\star]-[D'',\ol\tau^\star]=
[D'+\tau,\delta'+\tau^\star]+T_\omega.$$

When $\omega $ is K\"ahler, then $\tau=T_\omega=0$ and Lemma~1.6 a)
shows that $[D',\delta'']=0$. Together with the adjoint relation
$[D'',\delta']=0$, this equality implies
$$\Delta=\Delta'+\Delta''.\leqno(1.15)$$
When $\omega$ is not K\"ahler, Lemma~1.6 a) can be written 
$[D'+\tau,\delta'']=0$ and we obtain more generally
$$[D+\tau,\delta+\tau^\star]=\big[(D'+\tau)+D'',(\delta'+\tau^\star)+\delta''
\big]=\Delta'_\tau+\Delta''.$$

\claim{(1.16) Proposition} Set $\Delta_\tau=[D+\tau,\delta+\tau^\star]$.
Then $\Delta_\tau=\Delta'_\tau+\Delta''$.
\endclaim

\section{2}{Basic a Priori Inequality}
Let $(X,\omega)$ be a {\it compact} hermitian manifold, $\dim_\bC X=n$,
and $E$ a hermitian holomorphic vector bundle over $X$. For any section
$u\in \cC^\infty_{p,q}(X,E)$ we have $\Ll\Delta''u,u\Gg=\|D''u\|^2+\|\delta''u\|^2$
and the similar formula for $\Delta'_\tau$ gives $\Ll\Delta'_\tau u,u\Gg\ge 0$.
Theorem 1.4 implies therefore
$$\|D''u\|^2+\|\delta''u\|^2\ge\int_X\big(\langle[\ii\Theta(E),\Lambda]u,u\rangle+
\langle T_\omega u,u\rangle\big)dV.\leqno(2.1)$$
This inequality is known as the {\it Bochner-Kodaira-Nakano} inequality.
When $u$ is $\Delta''$-harmonic, we get in particular
$$\int_X\big(\langle[\ii\Theta(E),\Lambda]u,u\rangle+
\langle T_\omega u,u\rangle\big)dV\le 0.\leqno(2.2)$$
These basic a priori estimates are the starting point of all vanishing 
theorems. Observe that $~[\ii\Theta(E),\Lambda]+T_\omega~$ is a hermitian operator acting
pointwise on $~\Lambda^{p,q}T^\star X\otimes E$ (the hermitian property can
be seen from the fact that this operator coincides with $\Delta''-\Delta'_\tau$
on smooth sections). Using Hodge theory (Cor.~VI-11.2), we get:

\claim{(2.3) Corollary} If the hermitian operator $[\ii\Theta(E),\Lambda]+T_\omega$
is positive definite on $\Lambda^{p,q}T^\star X\otimes E$, then
$H^{p,q}(X,E)=0$.\qed
\endclaim

In some circumstances, one can improve Cor.~2.3 thanks to the
following ``analytic continuation lemma" due to \cite{Aronszajn 1957}:

\claim{(2.4) Lemma} Let $M$ be a connected $\cC^\infty$-manifold, $F$ a vector bundle
over $M$, and $P$ a second order elliptic differential operator acting
on $\cC^\infty(M,F)$. Then any section $\alpha\in\ker\,P$ vanishing on a 
non-empty open subset of $M$ vanishes identically on $M$.
\endclaim

\claim{(2.5) Corollary} Assume that $X$ is compact and connected. If
$$[\ii\Theta(E),\Lambda]+T_\omega\in\Herm\big(\Lambda^{p,q}T^\star X\otimes E\big)$$
is semi-positive on $X$ and positive definite in at least one point $x_0\in X$,
then we have $H^{p,q}(X,E)=0$.
\endclaim

\proof{Proof} By (2.2) every $\Delta''$-harmonic $(p,q)$-form $u$ must 
vanish in the neighborhood of $x_0$ where $[\ii\Theta(E),\Lambda]+T_\omega>0$,
thus $u\equiv 0$. Hodge theory implies $H^{p,q}(X,E)=0$.\qed
\endproof

\section{3}{Kodaira-Akizuki-Nakano Vanishing Theorem}
The main goal of vanishing theorems is to find natural geometric or
algebraic conditions on a bundle $E$ that will ensure that some
cohomology groups with values in $E$ vanish. In the next three sections,
we prove various vanishing theorems for cohomology groups of a hermitian
{\it line bundle} $E$ over a {\it compact} complex manifold $X$.

\claim{(3.1) Definition} A hermitian holomorphic line bundle $E$ on $X$
is said to be positive $($resp.\ negative$)$ if the hermitian matrix
$\big(c_{jk}(z)\big)$ of its Chern curvature form
$$\ii\Theta(E)=\ii\sum_{1\le j,k\le n}c_{jk}(z)\,dz_j\wedge d\ol z_k$$
is positive $($resp.\ negative$)$ definite at every point $z\in X$.
\endclaim

Assume that $X$ has a K\"ahler metric $\omega$.  Let
$$\gamma_1(x)\le\ldots\le\gamma_n(x)$$
be the eigenvalues of $\ii\Theta(E)_x$ with respect to $\omega_x$ at each point
$x\in X$, and let
$$\ii\Theta(E)_x=\ii\sum_{1\le j\le n}\gamma_j(x)\,\zeta_j\wedge\ol\zeta_j,
~~~~\zeta_j\in T^\star_x X$$
be a diagonalization of $\ii\Theta(E)_x$. By Prop.~VI-8.3 we have
$$\leqalignno{
\langle [\ii\Theta(E),\Lambda]u,u\rangle&=\sum_{J,K}
\Big(\sum_{j\in J}\gamma_j+\sum_{j\in K}\gamma_j
-\sum_{1\le j\le n}\gamma_j\Big)|u_{J,K}|^2\cr
&\ge(\gamma_1+\ldots+\gamma_q-\gamma_{p+1}-\ldots-\gamma_n)|u|^2
&(3.2)\cr}$$
for any form $u=\sum_{J,K}u_{J,K}\,\zeta_J\wedge\ol\zeta_K
\in\Lambda^{p,q}T^\star X$. 

\claim{(3.3) Akizuki-Nakano vanishing theorem (1954)} Let $E$
be a holomorphic line bundle on $X$.
\smallskip
\item{\rm a)} If $E$ is positive, then~ 
$H^{p,q}(X,E)=0~~$ for~ $p+q\ge n+1.$
\smallskip
\item{\rm b)} If $E$ is negative, then~
$H^{p,q}(X,E)=0~~$ for~ $p+q\le n-1.$
\endclaim

\proof{Proof} In case a), choose $\omega=\ii\Theta(E)$ as a K\"ahler metric on $X$.
Then we have $\gamma_j(x)=1$ for all $j$ and $x$, so that
$$\Ll [\ii\Theta(E),\Lambda]u,u\Gg\ge (p+q-n)||u||^2$$
for any $u\in\Lambda^{p,q}T^\star X\otimes E$. Assertion a) follows now from
Corollary~2.3. Property~b) is proved similarly, by taking
$\omega=-\ii\Theta(E)$. One can also derive b) from a) by Serre
duality (Theorem~VI-11.3).\qed
\endproof

When $p=0$ or $p=n$, Th.~3.3 can be generalized
to the case where $\ii\Theta(E)$ degenerates at some points.
We use here the standard notations
$$\Omega^p_X=\Lambda^pT^\star X,~~~~K_X=\Lambda^nT^\star X,~~~~
n=\dim_\bC X~;\leqno(3.4)$$
$K_X$ is called the {\it canonical line bundle} of $X$.

\claim{(3.5) Theorem (\cite{Grauert-Riemenschneider 1970})} Let $(X,\omega)$
be a compact and connected K\"ahler manifold and $E$ a line bundle on $X$.
\smallskip
\item{\rm a)} If $\ii\Theta(E)\ge 0$ on $X$ and $\ii\Theta(E)>0$ in at
least one point $x_0\in X$, then
\smallskip
\centerline{$H^q(X,K_X\otimes E)=0~~~~\hbox{\rm for}~~q\ge 1.$}
\smallskip
\item{\rm b)} If $\ii\Theta(E)\le 0$ on $X$ and $\ii\Theta(E)<0$ in at least 
one point $x_0\in X$, then
\smallskip
\centerline{$H^q(X,E)=0~~~~\hbox{\rm for}~~q\le n-1.$}
\endclaim

It will be proved in Volume II, by means of holomorphic Morse
inequalities, that the K\"ahler assumption is in fact unnecessary.
This improvement is a deep result first proved by \cite{Siu 1984} with
a different ad hoc method.

\proof{Proof} For $p=n$, formula (3.2) gives
$$\Ll [\ii\Theta(E),\Lambda]u,u\Gg\ge (\gamma_1+\ldots+\gamma_q)|u|^2\leqno(3.6)$$
and a) follows from Cor.~2.5. Now b) is a consequence of a) by
Serre duality.\qed
\endproof

\section{4}{Girbau's Vanishing Theorem}
Let $E$ be a line bundle over a compact connected K\"ahler manifold 
$(X,\omega)$. Girbau's theorem deals with the (possibly everywhere)
degenerate semi-positive case. We first state the corresponding
generalization of Th.~4.5.

\claim{(4.1) Theorem} If $\ii\Theta(E)$ is semi-positive and has at least
$n-s+1$ positive eigenvalues at a point $x_0\in X$ for some integer
$s\in\{1,\ldots,n\}$, then
$$H^q(X,K_X\otimes E)=0~~~~\hbox{\rm for}~~q\ge s.$$
\endclaim

\proof{Proof} Apply 2.5 and inequality (3.6), and observe
that $\gamma_q(x_0)>0$ for all $q\ge s$.\qed
\endproof

\claim{(4.2) Theorem {\rm (\cite{Girbau 1976})}} If $\ii\Theta(E)$ 
is semi-positive
and has at least $n-s+1$ positive eigenvalues at every point $x\in X$, then
$$H^{p,q}(X,E)=0~~~~\hbox{\rm for}~~p+q\ge n+s.$$
\endclaim

\proof{Proof} Let us consider on $X$ the new K\"ahler metric
$$\omega_\varepsilon=\varepsilon\omega +\ii\Theta(E),~~~~\varepsilon>0,$$
and let $\ii\Theta(E)=\ii\sum\gamma_j\,\zeta_j\wedge\ol\zeta_j$ be a diagonalization of
$\ii\Theta(E)$ with respect to $\omega$ and with $\gamma_1\le\ldots\le
\gamma_n$. Then
$$\omega_\varepsilon=\ii\sum~(\varepsilon+\gamma_j)\,\zeta_j\wedge\ol\zeta_j.$$
The eigenvalues of $\ii\Theta(E)$ with respect to $\omega_\varepsilon$ are given
therefore by 
$$\gamma_{j,\varepsilon}=\gamma_j/(\varepsilon+\gamma_j)\in[0,1[,
~~~~1\le j\le n.\leqno(4.3)$$
On the other hand, the hypothesis is equivalent to $\gamma_s>0$ on $X$.
For $j\ge s$ we have $\gamma_j\ge\gamma_s$, thus
$$\gamma_{j,\varepsilon}={1\over 1+\varepsilon/\gamma_j}\ge{1\over 1+
\varepsilon/\gamma_s}\ge1-\varepsilon/\gamma_s,~~~~s\le j\le n.~\leqno(4.4)$$
Let us denote the operators and inner products associated to 
$\omega_\varepsilon$ with $\varepsilon$ as an index. Then inequality
(3.2) combined with (4.4) implies
$$\eqalign{
\langle[\ii\Theta(E),\Lambda_\varepsilon]u,u\rangle_\varepsilon
&\ge\Big(\big(q-s+1)\big)(1-\varepsilon/\gamma_s)-(n-p)\Big)|u|^2\cr
&=\big(p+q-n-s+1-(q-s+1)\varepsilon/\gamma_s\big)|u|^2.\cr}$$
Theorem 4.2 follows now from Cor.~2.3 if we choose
$$\varepsilon<{p+q-n-s+1\over q-s+1}~\min_{x\in X}\gamma_s(x).\eqno{\square}$$
\endproof

\claim{(4.5) Remark} \rm The following example due to 
\cite{Ramanujam 1972, 1974} shows
that Girbau's result is no longer true for $p<n$ when $\ii\Theta(E)$ is only 
assumed to have $n-s+1$ positive eigenvalues on a dense open set.

Let $V$ be a hermitian vector space of dimension $n+1$ and $X$
the manifold obtained from $P(V)\simeq\bP^n$ by blowing-up
one point $a$. The manifold $X$ may be described
as follows: if $P(V/\bC a)$ is the projective space
of lines $\ell$ containing $a$, then
$$X=\big\{(x,\ell)\in P(V)\times P(V/\bC a)~;~x\in\ell\big\}.$$
We have two natural projections
$$\eqalign{\pi_1~:~&X\longrightarrow P(V)\simeq\bP^n,\cr
           \pi_2~:~&X\longrightarrow Y=P(V/\bC a)\simeq\bP^{n-1}.\cr}$$
It is clear that the preimage $\pi_1^{-1}(x)$ is the single point 
$\big(x,\ell=(a x)\big)$ if $x\ne a$ and that $\pi_1^{-1}(a)=\{a\}\times
Y\simeq\bP^{n-1}$, therefore
$$\pi_1~:~X\setminus(\{a\}\times Y)\longrightarrow P(V)\setminus\{a\}$$
is an isomorphism. On the other hand, $\pi_2$ is a locally trivial fiber
bundle over $Y$ with fiber $\pi_2^{-1}(\ell)=\ell\simeq \bP^1$,
in particular $X$ is smooth and $n$-dimensional. Consider now the
line bundle $E=\pi_1^\star\cO(1)$
over $X$, with the hermitian metric induced by that of $\cO(1)$. Then $E$
is semi-positive and $\ii\Theta(E)$ has $n$ positive eigenvalues at every point of
$X\setminus(\{a\}\times Y)$, hence the assumption of Th.~4.2 is
satisfied on $X\setminus(\{a\}\times Y)$. However, we will see that
$$H^{p,p}(X,E)\ne 0,~~~~0\le p\le n-1,$$
in contradiction with the expected generalization of (4.2) when 
\hbox{$2p\ge n+1$.} Let $j:Y\simeq\{a\}
\times Y\longrightarrow X$ be the inclusion. Then $\pi_1\circ j~:~Y\to\{a\}$ and
$\pi_2\circ j=\Id_Y\,;$ in particular $j^\star E=(\pi_1\circ j)^\star
\cO(1)$ is the trivial bundle $Y\times \cO(1)_a$. Consider now the
composite morphism
$$\cmalign{
H^{p,p}(Y,\bC)&\otimes H^0\big(P(V),\cO(1)\big)&\longrightarrow H^{p,p}(X,E)
{\buildo j^\star\over\longrightarrow}\,H^{p,p}(Y,\bC)\otimes\cO(1)_a\cr
\hfill u&\otimes s\hfill&\longmapsto \pi_2^\star u\otimes\pi_1^\star s,
\cr}$$
given by $u\otimes s\longmapsto(\pi_2\circ j)^\star u\otimes
(\pi_1\circ j)^\star s= u\otimes s(a)\,;$ it is surjective and
$H^{p,p}(Y,\bC)\ne 0$~ for $0\le p\le n-1$, so we have 
$H^{p,p}(X,E)\ne 0$.\qed
\endclaim

\section{5}{Vanishing Theorem for Partially Positive Line Bundles}
Even in the case when the curvature form $\ii\Theta(E)$ is not
semi-positive, some  cohomology groups of high tensor powers $E^k$
still vanish under suitable  assumptions. The prototype of such results
is the following assertion, which can be seen as a consequence of the
Andreotti-Grauert theorem \cite{Andreotti-Grauert 1962}, see IX-?.?; the
special case where $E$ is $>0$ (that is, $s=1$) is due to \cite{Kodaira
1953} and \cite{Serre 1956}.

\claim{(5.1) Theorem} Let $F$ be a holomorphic vector bundle over a compact
complex manifold $X$, $s$ a positive integer and $E$ a hermitian line
bundle such that $\ii\Theta(E)$ has at least $n-s+1$ positive eigenvalues at every
point $x\in X$. Then there exists an integer $k_0\ge 0$ such that
$$H^q(X,E^k\otimes F)=0~~~~\hbox{\rm for}~~q\ge s~~\hbox{\rm and}~~k\ge k_0.$$
\endclaim

\proof{Proof} The main idea is to construct a hermitian metric 
$\omega_\varepsilon$ on $X$ in such a way that all negative eigenvalues
of $\ii\Theta(E)$ with respect to $\omega_\varepsilon$ will be of small
absolute value. Let $\omega$ denote a fixed hermitian metric on $X$ and
let $\gamma_1\le\ldots\le\gamma_n$ be the corresponding eigenvalues of
$\ii\Theta(E)$.

\claim{(5.2) Lemma} Let $\psi\in \cC^\infty(\bR,\bR)$. If $A$ is a
hermitian $n\times n$ matrix with eigenvalues
$\lambda_1\le\ldots\le\lambda_n$ and corresponding eigenvectors $v_1,\ldots,
v_n$, we define $\psi[A]$ as the hermitian matrix with eigenvalues
$\psi(\lambda_j)$ and eigenvectors $v_j$, $1\le j\le n$. Then the map
$A\longmapsto\psi[A]$ is $\cC^\infty$ on $\Herm(\bC^n)$.
\endclaim

\proof{Proof} Although the result is very well known, we give here a short 
proof. Without loss of generality, we may assume that $\psi$ is compactly
supported. Then we have
$$\psi[A]={1\over 2\pi}\int_{-\infty}^{+\infty}\wh\psi(t)e^{itA}dt$$
where $\wh\psi$ is the rapidly decreasing Fourier transform of $\psi$.
The equality $\int_0^t(t-u)^pu^q\,du=p!\,q!/(p+q+1)!~$ and obvious
power series developments yield
$$D_A(e^{itA})\cdot B=i\int_0^t e^{\ii(t-u)A}\,B\,e^{iuA}du.$$
Since $e^{iuA}$ is unitary, we get $\|D_A(e^{itA})\|\le|t|$. A differentiation
under the integral sign and Leibniz' formula imply by induction on $k$ the
bound \hbox{$\|D_A^k(e^{itA})\|\le|t|^k$.} Hence $A\longmapsto\psi[A]$ is
smooth.\qed
\endproof

Let us consider now the positive numbers
$$t_0=\inf_X \gamma_s>0,~~~~M=\sup_X\max_j|\gamma_j|>0.$$
We select a function $\psi_\varepsilon\in \cC^\infty(\bR,\bR)$ such that 
$$\psi_\varepsilon(t)=t~~\hbox{\rm for}~t\ge t_0,
~~\psi_\varepsilon(t)\ge t~~\hbox{\rm for}~0\le t\le t_0,
~~\psi_\varepsilon(t)=M/\varepsilon~~\hbox{\rm for}~t\le 0.$$
By Lemma~5.2, $\omega_\varepsilon:=\psi_\varepsilon[\ii\Theta(E)]$ is a smooth
hermitian metric on $X$. Let us write
$$\ii\Theta(E)=\ii\sum_{1\le j\le n}\gamma_j\,\zeta_j\wedge\ol\zeta_j,~~~~
\omega_\varepsilon=\ii\sum_{1\le j\le n}\psi_\varepsilon(\gamma_j)\,
\zeta_j\wedge\ol\zeta_j$$
in an orthonormal basis $(\zeta_1,\ldots,\zeta_n)$ of $T^\star X$ for $\omega$.
The eigenvalues of $\ii\Theta(E)$ with respect to $\omega_\varepsilon$ are given by
$\gamma_{j,\varepsilon}=\gamma_j/\psi_\varepsilon(\gamma_j)$ and the
construction of $\psi_\varepsilon$ shows that $-\varepsilon\le
\gamma_{j,\varepsilon}\le1$, $1\le j\le n$, and $\gamma_{j,\varepsilon}=1$
for $s\le j\le n$. Now, we have
$$H^q(X,E^k\otimes F)\simeq H^{n,q}(X,E^k\otimes G)$$
where $G=F\otimes K^\star_X$. Let $e$, $(g_\lambda)_{1\le\lambda\le r}$
and $(\zeta_j)_{1\le j\le n}$ denote orthonormal frames of $E$, $G$ and
$(T^\star X,\omega_\varepsilon)$ respectively. For
$$u=\sum_{|J|=q,\lambda}u_{J,\lambda}\,\zeta_1\wedge\ldots\wedge\zeta_n
\wedge\ol\zeta_J\otimes e^k\otimes g_\lambda
\in\Lambda^{n,q}T^\star X\otimes E^k\otimes G,$$
inequality (3.2) yields
$$\langle[\ii\Theta(E),\Lambda_\varepsilon]u,u\rangle_\varepsilon=\sum_{J,\lambda}
\Big(\sum_{j\in J}\gamma_{j,\varepsilon}\Big)|u_{J,\lambda}|^2\ge
\big(q-s+1-(s-1)\varepsilon\big)|u|^2.$$
Choosing $\varepsilon=1/s$ and $q\ge s$, the right hand side becomes
$\ge (1/s)|u|^2$. Since $\Theta(E^k\otimes G)=k\Theta(E)\otimes\Id_G+\Theta(G)$,
there exists an integer $k_0$ such that
$$\big[\ii\Theta(E^k\otimes G),\Lambda_\varepsilon\big]+T_{\omega_\varepsilon}~~~~
\hbox{\rm acting~on~~}\Lambda^{n,q}T^\star X\otimes E^k\otimes G$$
is positive definite for $q\ge s$ and $k\ge k_0$. The proof is complete.\qed
\endproof

\section{6}{Positivity Concepts for Vector Bundles}
Let  $E$ be a hermitian holomorphic vector bundle of rank $r$ over $X$, where
$\dim_{\bC}X = n$. Denote by $(e_1,\ldots,e_r)$ an orthonormal frame of
$E$ over a coordinate patch $\Omega \subset X$ with complex coordinates
$(z_1,\ldots,z_n)$, and
$$\ii\Theta(E) = \ii\sum_{1\le j,k\le n,~1\le \lambda,\mu\le r}
c_{jk\lambda\mu}\,dz_j \wedge d\ol z_k \otimes e^\star_\lambda \otimes e_\mu,
~~~~\ol c_{jk\lambda\mu} = c_{kj\mu\lambda}
\leqno (6.1)$$
the Chern curvature tensor. To $\ii\Theta(E)$ corresponds a natural hermitian
form $\theta_E$ on $TX\otimes E$ defined by
$$\theta_E = \sum_{j,k,\lambda,\mu} c_{jk\lambda\mu}(dz_j \otimes
e^\star_\lambda) \otimes (\ol{dz_k \otimes e^\star_\mu}),$$
and such that
$$\theta_E(u,u) = \sum_{j,k,\lambda,\mu}c_{jk\lambda\mu}(x)\,u_{j\lambda}
\ol u_{k\mu},~~~~u\in T_xX\otimes E_x.(6.2)$$

\claim{(6.3) Definition {\rm (\cite{Nakano 1955})}} $E$ is said to be
Nakano positive $($resp.\ Nakano semi-negative$)$ if $\theta_E$ is
positive definite $($resp.\ semi-negative$)$ as a hermitian form on
$TX\otimes E$, i.e.\ if for every $u\in TX\otimes E,~u\ne 0,$ we have
$$\theta_E(u,u) > 0\quad(\hbox{\it resp.}\le 0).$$
We write $>_\Nak$~~$($resp.~$\le_\Nak)$ for Nakano positivity $($resp.\
semi-negativity$)$.
\endclaim

\claim{(6.4) Definition {\rm (\cite{Griffiths 1969})}} $E$ is said to
be Griffiths positive $($resp.\ Griffiths semi-negative$)$ if for all
$\xi \in T_xX$, $\xi\ne 0$ and $s\in E_x$, $s\ne 0$ we have
$$\theta_E(\xi\otimes s,\xi\otimes s) > 0\quad (\hbox{\it resp.}\le 0).$$
We write $>_\Grif$ $($resp.\ $\le_\Grif)$ for Griffiths positivity 
$($resp.\ semi-negativity$)$.
\endclaim

It is clear that Nakano positivity implies Griffiths positivity and that
both concepts coincide if $r=1$ (in the case of a line bundle, $E$ is merely
said to be positive). One can generalize further by introducing additional
concepts of positivity which interpolate between Griffiths positivity and
Nakano positivity.

\claim{(6.5) Definition} Let $T$ and $E$ be complex vector spaces of dimensions
$n,r$ respectively, and let $\Theta$ be a hermitian form on $T\otimes E$.
\smallskip
\item{\rm a)} A tensor $u\in T\otimes E$ is said to be of rank $m$ if $m$
is the smallest $\ge 0$ integer such that $u$ can be written
$$u = \sum_{j=1}^m \xi_j \otimes s_j,~~~~\xi_j\in T,~s_j\in E.$$
\smallskip
\item{\rm b)} $\Theta$ is said to be $m$-positive $($resp.\ 
$m$-semi-negative$)$ if
$\Theta(u,u)>0$ $($resp.\ $\Theta(u,u)\le 0)$ for every tensor $u\in T\otimes E$
of rank $\le m$, $u\ne 0$. In this case, we write
$$\Theta >_m 0~~~~(\hbox{\it resp.}~~\Theta \le_m 0).$$\smallskip
\endclaim

We say that the bundle  $E$ is $m$-positive if $\theta_E >_m 0$. Griffiths
positivity corresponds to $m=1$ and Nakano positivity to $m\ge \min(n,r)$.

\claim{(6.6) Proposition} A bundle $E$ is Griffiths positive if and only if
$E^\star$ is Griffiths negative.
\endclaim

\proof{Proof} By (V-4.$3'$) we get $\ii\Theta(E^\star) = -\ii\Theta(E)^\dagger$, hence
$$\theta_{E^\star}(\xi_1\otimes\ol s_2,\xi_2\otimes\ol s_1)=
-\theta_E(\xi_1\otimes s_1,\xi_2\otimes s_2),~~~~\forall\xi_1,\xi_2\in TX,~
\forall s_1,s_2\in E,$$
where $\ol s_j=\langle\bu,s_j\rangle\in E^\star$. Proposition 6.6 
follows immediately.\qed
\endproof

It should be observed that the corresponding duality property for Nakano
positive bundles is {\it not true}. In fact, using (6.1) we get
$$\ii\Theta(E^\star) = -\ii\sum_{j,k,\lambda,\mu} c_{jk\mu\lambda} dz_j\wedge d\ol z_k 
\otimes e^{\star\star}_\lambda \otimes e^\star_\mu,$$
$$\theta_{E^\star}(v,v)=-\sum_{j,k,\mu,\lambda}c_{jk\mu\lambda}v_{j\lambda}
\ol v_{k\mu},\leqno(6.7)$$
for any $v=\sum v_{j\lambda}\,(\partial/\partial z_j)\otimes e^\star_\lambda
\in TX\otimes E^\star$. The following example 
shows that Nakano positivity or negativity of $\theta_E$ and $\theta_{E^\star}$
are unrelated.

\claim{(6.8) Example} \rm Let $H$ be the rank $n$ bundle over $\bP^n$
defined in \S~V-15. For any $u=\sum u_{j\lambda}(\partial/\partial z_j)\otimes
\wt e_\lambda\in TX\otimes H$, $v=\sum v_{j\lambda}(\partial/\partial z_j)
\otimes\wt e_\lambda^\star\in TX\otimes H^\star$, 
$1\le j,\lambda\le n$, formula (V-15.9) implies
$$\left\{ \eqalign{
\theta_H(u,u)&=\sum u_{j\lambda}\ol u_{\lambda j}\cr
\theta_{H^\star}(v,v)&=\sum v_{jj}\ol v_{\lambda\lambda}=
\big|\sum v_{jj}\big|^2.\cr}\right.\leqno(6.9)$$
It is then clear that $H\ge_\Grif 0$ and $H^\star\le_\Nak 0$ , but
$H$ is neither $\ge_\Nak0$ nor $\le_\Nak0$.
\endclaim

\claim{(6.10) Proposition} Let $0\to S \to E \to Q \to 0$ be an exact
sequence of hermitian vector bundles. Then
\medskip
\item{\rm a)} $E\ge_\Grif 0~~\Longrightarrow~~Q\ge_\Grif 0,$
\smallskip
\item{\rm b)} $E\le_\Grif 0~~\Longrightarrow~~S\le_\Grif 0,$
\smallskip
\item{\rm c)} $E\le_\Nak0~~\Longrightarrow~~S\le_\Nak 0,$
\smallskip
\noindent and analogous implications hold true for strict positivity.
\endclaim

\proof{Proof} If $\beta$ is written $\sum dz_j \otimes \beta_j$,~$\beta_j 
\in \hom(S,Q)$, then formulas (V-14.6) and (V-14.7) yield
$$\eqalign{
\ii\Theta(S) &= \ii\Theta(E)_{\restriction S} - \sum dz_j \wedge d\ol z_k
\otimes \beta^\star_k \beta_j,\cr
\ii\Theta(Q) &= \ii\Theta(E)_{\restriction Q} + \sum dz_j \wedge d\ol z_k
\otimes \beta_j\beta^\star_k.\cr}$$
Since $\beta\cdot(\xi\otimes s)= \sum \xi_j\beta_j\cdot s$ and $\beta^\star
\cdot(\xi\otimes s)= \sum \ol \xi_k\beta^\star_k\cdot s$ we get
$$\theta_S(\xi\otimes s,\xi'\otimes s') = \theta_E(\xi\otimes s,\xi'\otimes s')
- \sum_{j,k} \xi_j \ol \xi'_k \langle \beta_j\cdot s,\beta_k\cdot s'\rangle,$$
$$\theta_S(u,u) = \theta_E(u,u) - |\beta\cdot u|^2,$$
$$\theta_Q(\xi\otimes s,\xi'\otimes s') = \theta_E(\xi\otimes s,\xi'\otimes s')
+ \sum_{j,k} \xi_j \ol \xi'_k \langle \beta^\star_k\cdot s,\beta^\star_j\cdot 
s'\rangle,$$
$$\theta_Q(\xi\otimes s,\xi\otimes s) = \theta_E(\xi\otimes s,\xi\otimes s)
+|\beta^\star\cdot(\xi\otimes s)|^2.\eqno{\square}$$
\endproof

Since $H$ is a quotient bundle of the trivial bundle $\soul V$,
Example~6.8 shows that $E\ge_\Nak 0$ {\it does not imply} 
$Q\ge_\Nak 0$.

\section{7}{Nakano Vanishing Theorem}
Let $(X,\omega)$ be a compact K\"ahler manifold, $\dim_\bC X=n$, and $E\longrightarrow X$
a hermitian vector bundle of rank $r$. We are going to compute explicitly
the hermitian operator $[\ii\Theta(E),\Lambda]$ acting on $\Lambda^{p,q}T^\star X
\otimes E$. Let $x_0\in X$ and $(z_1,\ldots,z_n)$ be local coordinates such that
$(\partial/\partial z_1,\ldots,\partial/\partial z_n)$ is an orthonormal
basis of $(TX,\omega)$ at $x_0$. One can write
$$\eqalign{
\omega_{x_0}&=\ii\sum_{1\le j\le n}dz_j\wedge d\ol z_j,\cr
\ii\Theta(E)_{x_0}&=\ii\sum_{j,k,\lambda,\mu}c_{jk\lambda\mu}\,dz_j\wedge d\ol z_k
\otimes e_\lambda^\star\otimes e_\mu\cr}$$
where $(e_1,\ldots,e_r)$ is an orthonormal basis of $E_{x_0}$. Let
$$u=\sum_{|J|=p,\,|K|=q,\,\lambda}u_{J,K,\lambda}\,dz_J\wedge d\ol z_K
\otimes e_\lambda\in\big(\Lambda^{p,q}T^\star X\otimes E\big)_{x_0}.$$
A simple computation as in the proof of Prop.~VI-8.3 gives
$$\eqalign{
\Lambda u&=\ii(-1)^p\sum_{J,K,\lambda,s}u_{J,K,\lambda}\,
\Big({\partial\over\partial z_s}\ort dz_J\Big)\wedge 
\Big({\partial\over\partial\ol z_s}\ort d\ol z_K\Big)\otimes e_\lambda,\cr
\ii\Theta(E)\wedge u&=\ii(-1)^p\sum_{j,k,\lambda,\mu,J,K}c_{jk\lambda\mu}\,
u_{J,K,\lambda}\,dz_j\wedge dz_J\wedge d\ol z_k\wedge d\ol z_K\otimes e_\mu,\cr
[\ii\Theta(E),\Lambda]u&=\sum_{j,k,\lambda,\mu,J,K}
c_{jk\lambda\mu}\,u_{J,K,\lambda}\,dz_j\wedge\Big({\partial\over\partial z_k}
\ort dz_J\Big)\wedge d\ol z_K\otimes e_\mu\cr
&+\sum_{j,k,\lambda,\mu,J,K}c_{jk\lambda\mu}\,u_{J,K,\lambda}\,dz_J\wedge d\ol 
z_k\wedge\Big({\partial\over\partial\ol z_j}\ort d\ol z_K\Big)\otimes e_\mu\cr
&-\sum_{j,\lambda,\mu,J,K}c_{jj\lambda\mu}\,u_{J,K,\lambda}\,dz_J\wedge d\ol z_K
\otimes e_\mu.\cr}$$
We extend the definition of $u_{J,K,\lambda}$ to non increasing multi-indices
$J=(j_s)$, $K=(k_s)$ by deciding that $u_{J,K,\lambda}=0$ if $J$ or $K$ 
contains identical components repeated and that $u_{J,K,\lambda}$ is 
alternate in the indices $(j_s)$, $(k_s)$. Then the above equality 
can be written
$$\eqalign{
\langle[\ii\Theta(E),\Lambda]u,u\rangle&=
\sum c_{jk\lambda\mu}\,u_{J,jS,\lambda}{\ol u_{J,kS,\mu}}\cr
&+\sum c_{jk\lambda\mu}\,u_{kR,K,\lambda}{\ol u_{jR,K,\mu}}\cr
&-\sum c_{jj\lambda\mu}\,u_{J,K,\lambda}{\ol u_{J,K,\mu}},\cr}$$
extended over all indices $j,k,\lambda,\mu,J,K,R,S$ with 
$|R|=p-1$, $|S|=q-1$. This hermitian form appears rather 
difficult to handle for general $(p,q)$ because of sign compensation. 
Two interesting cases are $p=n$ and $q=n$.
\medskip
\noindent$\bu$
For $u=\sum u_{K,\lambda}\,dz_1\wedge\ldots\wedge dz_n\wedge d\ol z_K\otimes
e_\lambda$ of type $(n,q)$, we get
$$\langle[\ii\Theta(E),\Lambda]u,u\rangle=\sum_{|S|=q-1}\sum_{j,k,\lambda,\mu}
c_{jk\lambda\mu}\,u_{jS,\lambda}\ol u_{kS,\mu},\leqno(7.1)$$
because of the equality of the second and third summations in the general
formula. Since $u_{jS,\lambda}=0$ for $j\in S$, the rank of the tensor
$(u_{jS,\lambda})_{j,\lambda}\in\bC^n\otimes\bC^r$ is in fact 
$\le\min\{n-q+1,r\}$. We obtain therefore:

\claim{(7.2) Lemma} Assume that $E>_m0$ in the sense of Def.~$6.5$. Then the
hermitian operator $[\ii\Theta(E),\Lambda]$ is positive definite on
$\Lambda^{n,q}T^\star X\otimes E$ for $q\ge 1$ and $m\ge\min\{n-q+1,r\}.$
\endclaim

\claim{(7.3) Theorem} Let $X$ be a compact connected K\"ahler manifold of
dimension $n$ and $E$ a hermitian vector bundle of rank $r$.
If $\theta_E\ge_m 0$ on $X$ and $\theta_E>_m0$ in at least one point, then
$$H^{n,q}(X,E)=H^q(X,K_X\otimes E)=0~~~~\hbox{\rm for}~~q\ge 1~~\hbox{\rm and}~~
m\ge\min\{n-q+1,r\}.$$
\endclaim

\vskip-6pt\noindent$\bu$
Similarly, for $u=\sum u_{J,\lambda}\,dz_J\wedge d\ol z_1\wedge\ldots\wedge 
d\ol z_n\otimes e_\lambda$ of type $(p,n)$, we get
$$\langle[\ii\Theta(E),\Lambda]u,u\rangle=\sum_{|R|=p-1}\sum_{j,k,\lambda,\mu}
c_{jk\lambda\mu}\,u_{kR,\lambda}\ol u_{jR,\mu},$$
because of the equality of the first and third summations in the general
formula. The indices $j,k$ are twisted, thus $[\ii\Theta(E),\Lambda]$
defines a positive hermitian form under the assumption
$\ii\Theta(E)^\dagger>_m 0$, i.e.\ $\ii\Theta(E^\star)<_m 0$, with
\hbox{$m\ge\min\{n-p+1,r\}$.} Serre duality
$\big(H^{p,0}(X,E)\big)^\star=H^{n-p,n}(X,E^\star)$~gives:

\claim{(7.4) Theorem} Let $X$ and $E$ be as above.
If $\theta_E\le_m 0$ on $X$ and $\theta_E<_m 0$ in at least one point, then
$$H^{p,0}(X,E)=H^0(X,\Omega^p_X\otimes E)=0~~~~\hbox{\rm for}~~p<n~~\hbox{\rm and}~~
m\ge\min\{p+1,r\}.$$
\endclaim

\vskip-6pt\noindent The special case $m=r$ yields:

\claim{(7.5) Corollary} For $X$ and $E$ as above:
\smallskip
\item{\rm a)} Nakano vanishing theorem (1955):
\smallskip
\item{~} $E\ge_\Nak0,~~~\hbox{\it strictly in one point}~~~
\Longrightarrow~~~H^{n,q}(X,E)=0~~~\hbox{\it for}~~q\ge 1.$
\smallskip
\item{\rm b)} $E\le_\Nak0$,~~~~strictly in one point~~~
$\Longrightarrow~~~H^{p,0}(X,E)=0$~~~for~~$p<n$.
\endclaim

\section{8}{Relations Between Nakano and Griffiths Positivity}
It is clear that Nakano positivity implies Griffiths positivity. The
main result of \S~8 is the following ``converse'' to this property 
\cite{Demailly-Skoda 1979}.

\claim{(8.1) Theorem} For any hermitian vector bundle $E$, 
$$E >_\Grif 0~~\Longrightarrow~~E\otimes \det E >_\Nak 0.$$
\endclaim

To prove this result, we first use (V-4.$2'$) and (V-4.6). If 
$\End(E\otimes\det E)$ is identified to $\hom(E,E)$, one can write
$$\Theta(E\otimes \det E) = \Theta(E)+\Tr_E(\Theta(E)) \otimes \Id_E,$$
$$\theta_{E\otimes \det E} = \theta_E+\Tr_E\theta_E \otimes h,$$
where $h$ denotes the hermitian metric on $E$ and where $\Tr_E\theta_E$ is the
hermitian form on $TX$ defined by
$$\Tr_E \theta_E(\xi,\xi) = \sum_{1\le \lambda\le r} \theta_E(\xi\otimes
e_\lambda,\xi\otimes e_\lambda),~\xi\in TX,$$
for any orthonormal frame $(e_1,\ldots,e_r)$ of $E$. Theorem 8.1 is now a
consequence of the following simple property of hermitian forms on a tensor
product of complex vector spaces.

\claim{(8.2) Proposition} Let $T,E$ be complex vector spaces of respective
dimensions $n,r,\,$ and $h$ a hermitian metric on $E$. Then for
every hermitian form $\Theta$ on $T\otimes E$
$$\Theta >_\Grif 0~~\Longrightarrow~~
\Theta + \Tr_E\Theta \otimes h >_\Nak 0.$$
\endclaim

We first need a lemma analogous to Fourier inversion formula for discrete
Fourier transforms.

\claim{(8.3) Lemma} Let $q$ be an integer $\ge 3$, and $x_\lambda,~y_\mu,~1\le \lambda,\mu \le r$, be complex numbers. Let $\sigma$ describe
the set $U^r_q$ of $r$-tuples of $q$-th roots of unity and put
$$x'_\sigma = \sum_{1\le \lambda\le r} x_\lambda \ol \sigma_\lambda,~~~~
y'_\sigma = \sum_{1\le \mu\le r} y_\mu \ol \sigma_\mu,~~~~ \sigma\in U^r_q.$$
Then for every pair $(\alpha,\beta),~1\le \alpha,\beta\le r$, the following 
identity holds:
$$q^{-r} \sum_{\sigma\in U^r_q} x'_\sigma \ol y'_\sigma \sigma_\alpha \ol 
\sigma_\beta=\cases{
x_\alpha \ol y_\beta&if~~$\alpha\ne \beta,$\cr
\displaystyle\sum_{1\le \mu\le r} x_\mu \ol y_\mu&if~~$\alpha=\beta.$\cr}$$
\endclaim

\proof{Proof} The coefficient of $x_\lambda\ol y_\mu$ in the summation
$q^{-r} \sum_{\sigma\in U^r_q} x'_\sigma \ol y'_\sigma \sigma_\alpha \ol 
\sigma_\beta$ is given by
$$q^{-r} \sum_{\sigma\in U^r_q}\sigma_\alpha \ol \sigma_\beta\ol
\sigma_\lambda \sigma_\mu.$$
This coefficient equals 1 when the pairs $\{ \alpha,\mu\}$ and
$\{\beta,\lambda\}$ are equal (in which case $\sigma_\alpha
\ol \sigma_\beta \ol \sigma_\lambda \sigma_\mu = 1$ for any one of the $q^r$
elements of $U^r_q$). Hence, it is sufficient to prove that
$$\sum_{\sigma\in U^r_q} \sigma_\alpha \ol \sigma_\beta \ol \sigma_\lambda
\sigma_\mu = 0$$
when the pairs $\{ \alpha,\mu\}$ and $\{ \beta,\lambda\}$
are distinct.

If $\{ \alpha,\mu\} \ne \{\beta,\lambda\}$, then one of the
elements of one of the pairs does not belong to the other pair. As the four
indices $\alpha,\beta,\lambda,\mu$ play the same role, we may suppose for 
example that $\alpha\notin \{\beta,\lambda\}$. Let us apply to 
$\sigma$ the substitution $\sigma\mapsto \tau$, where $\tau$ is defined by
$$\tau_\alpha = e^{2\pi\ii/q}\sigma_\alpha,~\tau_\nu = 
\sigma_\nu \quad \hbox{\rm for}\quad \nu\ne \alpha.$$
We get
$$\sum_\sigma\sigma_\alpha\ol\sigma_\beta\ol\sigma_\lambda\sigma_\mu=\sum_\tau
=\cases{
\displaystyle e^{2\pi\ii/q}\sum_\sigma&if~~$\alpha\ne\mu,$\cr
\displaystyle e^{4\pi\ii/q}\sum_\sigma&if~~$\alpha=\mu,$\cr}$$
Since $q\ge 3$ by hypothesis, it follows that
$$\sum_\sigma\sigma_\alpha\ol\sigma_\beta\ol\sigma_\lambda\sigma_\mu=0.$$
\endproof

\proof{Proof of Proposition 8.2.} Let $(t_j)_{1\le j\le n}$ be a basis of
$T$, $(e_\lambda)_{1\le \lambda\le r}$ an orthonormal basis of $E$ and
$\xi=\sum_j\xi_jt_j\in T$, $u=\sum_{j,\lambda}u_{j\lambda}\,t_j\otimes 
e_\lambda \in T\otimes E$. The coefficients $c_{jk\lambda\mu}$ of $\Theta$
with respect to the basis $t_j\otimes e_\lambda$ satisfy the symmetry
relation $\ol c_{jk\lambda\mu}=c_{kj\mu\lambda}$, and we have the formulas 
$$\eqalign{\Theta(u,u)
&=\sum_{j,k,\lambda,\mu}c_{jk\lambda\mu}u_{j\lambda}\ol u_{k\mu},\cr
\Tr_E\Theta(\xi,\xi)&=\sum_{j,k,\lambda}c_{jk\lambda\lambda}\xi_j\ol\xi_k,\cr
(\Theta+\Tr_E\Theta\otimes h)(u,u)&=\sum_{j,k,\lambda,\mu}c_{jk\lambda\mu} 
u_{j\lambda} \ol u_{k\mu}+ c_{jk\lambda\lambda}u_{j\mu}\ol u_{k\mu}.\cr}$$
For every $\sigma\in U^r_q$ (cf.\ Lemma~8.3), put
$$\eqalign{
u'_{j\sigma}&=\sum_{1\le\lambda\le r}u_{j\lambda}\ol\sigma_\lambda\in\bC,\cr
\wh u_\sigma&=\sum_j u'_{j\sigma}t_j\in T\quad,\quad
\wh e_\sigma=\sum_\lambda\sigma_\lambda e_\lambda\in E.\cr}$$
Lemma 8.3 implies
$$\eqalign{
q^{-r} \sum_{\sigma\in U^r_q}\Theta(\wh u_\sigma\otimes\wh e_\sigma,
\wh u_\sigma\otimes\wh e_\sigma)&= q^{-r} \sum_{\sigma\in U^r_q} c_{jk\lambda\mu}
u'_{j\sigma} \ol u'_{k\sigma}\sigma_\lambda \ol \sigma_\mu\cr
&= \sum_{j,k,\lambda\ne \mu} c_{jk\lambda\mu} u_{j\lambda} \ol u_{k\mu} +
\sum_{j,k,\lambda,\mu} c_{jk\lambda\lambda} u_{j\mu} \ol u_{k\mu}.\cr}$$
The Griffiths positivity assumption shows that the left hand side is $\ge 0$,
hence
$$(\Theta + \Tr_E \Theta \otimes h)(u,u) \ge \sum_{j,k,\lambda} c_{jk\lambda\lambda} u_{j\lambda} \ol u_{k\lambda} \ge 0$$
with strict positivity if $\Theta >_\Grif 0$ and $u\ne 0$.\qed
\endproof

\claim{(8.4) Example} \rm Take $E=H$ over $\bP^n=P(V)$. The exact sequence
$$0 \longrightarrow \cO(-1) \longrightarrow \soul V \longrightarrow H \longrightarrow 0$$
implies $\det \soul V=\det H \otimes \cO(-1)$. Since $\det \soul V$  
is a trivial bundle, we get (non canonical) isomorphisms
$$\eqalign{
\det H &\simeq \cO(1),\cr 
T\bP^n &= H\otimes \cO(1) \simeq H\otimes \det H.\cr}$$
We already know that $H\ge_\Grif 0$, hence $T\bP^n \ge_\Nak 0$.
A direct computation based on (6.9) shows that
$$\eqalign{
\theta_{T\bP^n}(u,u) &= (\theta_H + \Tr_H \theta_H \otimes h)(u,u)\cr
&= \sum_{1\le j,k\le n} u_{jk} \ol u_{kj} + u_{jk} \ol u_{jk}
 = {1\over 2} \sum_{1\le j,k\le n} |u_{jk} + u_{kj}|^2.\cr}$$
In addition, we have $T\bP^n>_\Grif 0$. However,
the Serre duality theorem gives
$$\eqalign{
H^q(\bP^n,K_{\bP^n}\otimes T\bP^n)^\star
&\simeq H^{n-q}(\bP^n,T^\star\bP^n)\cr
&=H^{1,n-q}(\bP^n,\bC)=\cases{
   \bC&if~~$q=n-1$,\cr
   0    &if~~$q\ne n-1$.\cr}\cr}$$
For $n\ge 2$, Th.~7.3 implies that $T\bP^n$ has no hermitian metric 
such that $\theta_{T\bP^n}\ge_20$ on $\bP^n$ and $\theta_{T\bP^n}>_20$ in 
one point. This shows that the notion of $2$-positivity is actually
stronger than $1$-positivity (i.e.\ Griffiths positivity).
\endclaim

\claim{(8.5) Remark} \rm Since $\Tr_H\theta_H = \theta_{\cO(1)}$ is positive
and $\theta_{T\bP^n}$ is not $>_\Nak 0$ when $n\ge 2$,
we see that Prop.~8.2 is best possible in the sense that there 
cannot exist any constant $c<1$ such that
$$\Theta>_\Grif0~~~\Longrightarrow~~~\Theta + c \Tr_E \Theta\otimes h
\ge_\Nak 0.$$
\endclaim

\section{9}{Applications to Griffiths Positive Bundles}
We first need a preliminary result.

\claim{(9.1) Proposition} Let $T$ be a complex vector space and
$(E,h)$ a hermitian vector space of respective dimensions $n,r$ with
$r\ge 2$. Then for any hermitian form $\Theta$ on $T\otimes E$ and any
integer $m\ge 1$
$$\Theta>_\Grif0~~~\Longrightarrow~~~m\Tr_E\Theta\otimes h-\Theta>_m0.$$
\endclaim

\proof{Proof} Let us distinguish two cases.
\medskip
\noindent a)~ $m=1$.~
Let $u\in T \otimes E$ be a tensor of rank 1. Then $u$ can be written
$u = \xi_1 \otimes e_1$ with $\xi_1\in T,~\xi_1 \ne 0$, and
$e_1 \in E,~|e_1| = 1$. Complete $e_1$ into an orthonormal basis
$(e_1,\ldots,e_r)$ of $E$. One gets immediately
$$\eqalign{
(\Tr_E \Theta \otimes h)(u,u) &= \Tr_E \Theta (\xi_1,\xi_1) = \sum_{1\le \lambda\le r} \Theta(\xi_1\otimes e_\lambda , \xi_1 \otimes e_\lambda)\cr
&> \Theta (\xi_1\otimes e_1,\xi_1\otimes e_1) = \Theta (u,u).\cr}$$
\medskip
\noindent b)~ $m\ge 2$.~
Every tensor $u\in T\otimes E$ of $\hbox{\rm rank~} \le m$ can be written
$$u=\sum_{1\le \lambda\le q} \xi_\lambda \otimes e_\lambda\quad,
\quad \xi_\lambda \in T,$$
with $q = \min(m,r)$ and $(e_\lambda)_{1\le \lambda\le r}$ an orthonormal basis
of $E$. Let $F$ be the vector subspace of $E$ generated by $(e_1,\ldots,e_q)$
and $\Theta_F$ the restriction of $\Theta$ to $T\otimes F$. The first part
shows that
$$\Theta':= \Tr_F \Theta_F \otimes h - \Theta_F >_\Grif 0.$$
Proposition 9.2 applied to $\Theta'$ on $T\otimes F$ yields
$$\Theta'+\Tr_F \Theta'\otimes h=q\Tr_F\Theta_F\otimes h-\Theta_F>_q 0.$$
Since $u\in T\otimes F$ is of $\hbox{\rm rank~} \le q \le m$, we get
(for $u\ne 0$)
$$\eqalignno{
\Theta(u,u) = \Theta_F(u,u) &< q(\Tr_F \Theta_F \otimes h)(u,u)\cr
&=q\sum_{1\le j,\lambda\le q}\Theta(\xi_j\otimes e_\lambda,
\xi_j\otimes e_\lambda)\le m\Tr_E\Theta\otimes h(u,u).&\square\cr}$$
\endproof

Proposition 9.1 is of course also true in the semi-positive case. From these
facts, we deduce

\claim{(9.2) Theorem} Let $E$ be a Griffiths $($semi-$)$positive bundle of 
rank $r\ge 2$. Then for any integer $m\ge 1$
$$E^\star \otimes (\det E)^m >_m 0~~~~(\hbox{\it resp.}~~\ge_m 0).$$
\endclaim

\proof{Proof} Apply Prop.~8.1 to $\Theta = -\theta_{E^\star} >_\Grif 0$ and observe that
$$\theta_{\det E} = - \theta_{\det E^\star} = \Tr_{E^\star} \Theta.$$
\endproof

\claim{(9.3) Theorem} Let $0\to S \to E \to Q \to 0$ be an exact sequence of 
hermitian vector bundles. Then for any $m\ge 1$
$$E>_m0~~~\Longrightarrow~~~S\otimes (\det Q)^m >_m 0.$$
\endclaim

\proof{Proof} Formulas (V-14.6) and (V-14.7) imply
$$\ii\Theta(S) >_m \ii\beta^\star \wedge \beta\quad,\quad \ii\Theta(Q) >_m \ii\beta \wedge \beta^\star,$$
$$\ii\Theta(\det Q) = \Tr_Q(\ii\Theta(Q)) > \Tr_Q(\ii\beta \wedge \beta^\star).$$
If we write $\beta = \sum dz_j \otimes \beta_j$ as in the proof of
Prop.~6.10, then
$$\eqalign{ \Tr_Q(\ii\beta \wedge \beta^\star) &= \sum idz_j \wedge d\ol
z_k \Tr_Q(\beta_j\beta^\star_k)\cr &=\sum idz_j \wedge d\ol z_k
\Tr_S(\beta^\star_k\beta_j) = \Tr_S(-\ii\beta^\star \wedge \beta).\cr}$$
Furthermore, it has been already proved that $-\ii\beta^\star \wedge
\beta \ge_\Nak 0$. By Prop.~8.1 applied to the corresponding
hermitian form $\Theta$ on $TX \otimes S$, we get $$m
\Tr_S(-\ii\beta^\star \wedge \beta) \otimes \Id_S + \ii\beta^\star\wedge
\beta \ge_m 0,$$ and Th.~9.3 follows.
\endproof

\claim{(9.4) Corollary} Let $X$ be a compact $n$-dimensional complex 
manifold, $E$ a vector bundle of rank $r\ge 2$ and $m\ge 1$ an integer. Then
\smallskip
\item{\rm a)} $E>_\Grif0 \Longrightarrow H^{n,q}(X,E\otimes
\det\,E)=0$~~~for~~$q\ge 1\,;$
\smallskip
\item{\rm b)} $E>_\Grif0\Longrightarrow H^{n,q}\big
(X,E^\star\otimes(\det\,E)^m\big)=0$~~~for~~$q\ge 1$\Fbreak
and~~$m\ge\min\{n-q+1,r\}\,;$
\smallskip
\item{\rm c)} Let~ $0\to S\to E\to Q\to 0$ be an exact sequence of
vector bundles and $m=\min\{n-q+1,\rk\,S\}$, $q\ge 1$. If $E>_m0$ and if
$L$ is a line bundle such that $L\otimes(\det Q)^{-m}\ge 0$, then
$$H^{n,q}(X,S\otimes L)=0.$$
\endclaim

\proof{Proof} Immediate consequence of Theorems 7.3, 8.1, 9.2 and 9.3.\qed
\endproof

Note that under our hypotheses $\omega=\ii\,\Tr_E\,\Theta(E)=
\ii\Theta(\Lambda^rE)$ is
a K\"ahler metric on $X$.
Corollary 2.5 shows that it is enough in a), b), c) to assume
semi-positivity and strict positivity in one point (this is true a priori
only if $X$ is supposed in addition to be K\"ahler, but this hypothesis
can be removed by means of Siu's result mentioned after~(4.5).

a) is in fact a special case of a result of \cite{Griffiths 1969}, which we
will prove in full generality in volume II (see the chapter on vanishing
theorems for ample vector bundles); property b) will be also considerably
strengthened there. Property c) is due to \cite{Skoda 1978} for $q=0$ and
to \cite{Demailly 1982c} in general. Let us take the 
tensor product of the exact sequence in c) with $(\det\,Q)^l$. The
corresponding long cohomology exact sequence implies that the natural
morphism
$$H^{n,q}\big(X,E\otimes(\det\,Q)^l\big)\longrightarrow 
  H^{n,q}\big(X,Q\otimes(\det\,Q)^l\big)$$
is surjective for $q\ge 0$ and $l,m\ge\min\{n-q,\rk\,S\}$, bijective 
for $q\ge 1$ and $l,m\ge\min\{n-q+1,\rk\,S\}$.

\section{10}{Cohomology Groups of $\cO(k)$ over $\bP^n$}
As an illustration of the above results, we compute now the cohomology groups
of all line bundles $\cO(k)\to\bP^n$. This precise evaluation will be needed
in the proof of a general vanishing theorem for vector bundles, due to Le
Potier (see volume II). As in \S V-15, we consider a complex vector space 
$V$ of dimension $n+1$ and the exact sequence
$$0\longrightarrow\cO(-1)\longrightarrow\soul V\longrightarrow H\longrightarrow 0\leqno(10.1)$$
of vector bundles over $\bP^n=P(V)$. We thus have \hbox{
$\det\soul V=\det H\otimes\cO(-1)$}, and as $TP(V)=H\otimes\cO(1)$ by
Th.~V-15.7, we find
$$K_{P(V)}=\det T^\star P(V)=\det H^\star\otimes\cO(-n)=\det\soul V^\star
\otimes\cO(-n-1)\leqno(10.2)$$
where~ $\det\soul V$ is a trivial line bundle.

Before going further, we need some notations. For every integer
$k\in\bN$, we consider the homological complex $C^{\bu,k}
(V^\star)$ with differential $\gamma$ such that
$$\left\{\eqalign{
C^{p,k}(V^\star)
=&~\Lambda^pV^\star\otimes S^{k-p}V^\star,~~~~0\le p\le k,\cr
=&~0~~~~~\hbox{\rm otherwise},\cr
\gamma~:~~\Lambda^pV^\star&\otimes S^{k-p}V^\star\longrightarrow\Lambda^{p-1}V^\star
\otimes S^{k-p+1}V^\star,\cr}\right.\leqno(10.3)$$
where $\gamma$ is the linear map obtained by contraction with the Euler vector
field $\Id_V\in V\otimes V^\star$, through the obvious maps
$V\otimes\Lambda^pV^\star\longrightarrow\Lambda^{p-1}V^\star$ and $V^\star\otimes
S^{k-p}V^\star\longrightarrow S^{k-p+1}V^\star$. If $(z_0,\ldots,z_n)$ are coordinates 
on $V$, the module $C^{p,k}(V^\star)$ can be identified with the space
of $p$-forms
$$\alpha(z)=\sum_{|I|=p}\alpha_I(z)\,dz_I$$
where the $\alpha_I$ 's are homogeneous polynomials of degree $k-p$.
The differential $\gamma$ is given by contraction with the Euler
vector field $\xi=\sum_{0\le j\le n}z_j\,\partial/\partial z_j$.

Let us denote by $Z^{p,k}(V^\star)$ the space of $p$-cycles of
$C^{\bu,k}(V^\star)$, i.e.\ the space of forms $\alpha\in
C^{p,k}(V^\star)$ such that $\xi\ort\alpha=0$. The exterior derivative
$d$ also acts on $C^{\bu,k}(V^\star)\,;$ we have
$$d~:~~C^{p,k}(V^\star)\longrightarrow C^{p+1,k}(V^\star),$$
and a trivial computation shows that $d\gamma+\gamma d=k\cdot\Id_
{C^{\bu,k}(V^\star)}.$

\claim{(10.4) Theorem} For $k\ne 0$, $C^{\bu,k}(V^\star)$ is exact and 
there exist canonical isomorphisms
$$C^{\bu,k}(V^\star)=\Lambda^pV^\star\otimes S^{k-p}V^\star\simeq
Z^{p,k}(V^\star)\oplus Z^{p-1,k}(V^\star).$$
\endclaim

\proof{Proof} The identity $d\gamma+\gamma d=k\cdot\Id$ implies
the exactness. The isomorphism is given by
${1\over k}\gamma d\oplus\gamma$ and its inverse by
$\cP_1+{1\over k}d\circ\cP_2$.\qed
\endproof

Let us consider now the canonical mappings
$$\pi~:~~V\setminus\{0\}\longrightarrow P(V),~~~~\mu'~:~~V\setminus\{0\}\longrightarrow\cO(-1)$$
defined in \S V-15. As $T_{[z]}P(V)\simeq V/\bC\xi(z)$ for
all $z\in V\setminus\{0\}$, every form $\alpha\in Z^{p,k}(V^\star)$
defines a holomorphic section of $\pi^\star\big(\Lambda^pT^\star P(V)\big)$,
$\alpha(z)$ being homogeneous of degree $k$ with respect to $z$.
Hence $\alpha(z)\otimes\mu'(z)^{-k}$ is a holomorphic section of
$\pi^\star\big(\Lambda^pT^\star P(V)\otimes\cO(k)\big)$, and since its
homogeneity degree is $0$, it induces a holomorphic section of
$\Lambda^pT^\star P(V)\otimes\cO(k)$. We thus have an injective morphism
$$Z^{p,k}(V^\star)\longrightarrow H^{p,0}\big(P(V),\cO(k)\big).\leqno(10.5)$$

\claim{(10.6) Theorem} The groups $H^{p,0}\big(P(V),\cO(k)\big)$ are given by
\smallskip
\item{\rm a)} $H^{p,0}\big(P(V),\cO(k)\big)\simeq Z^{p,k}(V^\star)$
~~for~~$k\ge p\ge 0,$
\smallskip
\item{\rm b)} $H^{p,0}\big(P(V),\cO(k)\big)=0$
~~for~~$k\le p$~~and~~$(k,p)\ne(0,0)$.
\endclaim

\proof{Proof} Let $s$ be a holomorphic section of $\Lambda^pT^\star P(V)
\otimes\cO(k)$. Set
$$\alpha(z)=(d\pi_z)^\star\big(s([z])\otimes\mu'(z)^k\big),~~~~z\in 
V\setminus\{0\}.$$
Then $\alpha$ is a holomorphic $p$-form on $V\setminus\{0\}$ such that
$\xi\ort\alpha=0$, and the coefficients of $\alpha$ are homogeneous of
degree $k-p$ on $V\setminus\{0\}$ (recall that $d\pi_{\lambda z}=\lambda^{-1}
d\pi_z$). It follows that $\alpha=0$ if $k<p$ and that 
$\alpha\in Z^{p,k}(V^\star)$ if $k\ge p$. The injective morphism (10.5)
is therefore also surjective. Finally, $Z^{p,p}(V^\star)=0$ for
$p=k\ne 0$, because of the exactness of $C^{\bu,k}(V^\star)$ when
$k\ne 0$. The proof is complete.\qed
\endproof

\claim{(10.7) Theorem} The cohomology groups $H^{p,q}\big(P(V),\cO(k)\big)$
vanish in the following cases:
\smallskip
\item{\rm a)} $q\ne 0,n,p\,;$
\smallskip
\item{\rm b)} $q=0,~~k\le p$~~and~~$(k,p)\ne(0,0)\,;$
\smallskip
\item{\rm c)} $q=n,~~k\ge-n+p$~~and~~$(k,p)\ne(0,n)\,;$
\smallskip
\item{\rm d)} $q=p\ne 0,n,~~k\ne 0.$
\medskip
\noindent The remaining non vanishing groups are:
\smallskip
\item{${\rm \ol b)}$} $H^{p,0}\big(P(V),\cO(k)\big)\simeq Z^{p,k}(V^\star)$
~~for~~$k>p\,;$
\smallskip
\item{${\rm \ol c)}$} $H^{p,n}\big(P(V),\cO(k)\big)\simeq Z^{n-p,-k}(V)$
~~for~~$k<-n+p\,;$
\smallskip
\item{${\rm \ol d)}$} $H^{p,p}\big(P(V),\bC\big)=\bC,~~~~0\le p\le n.$
\endclaim

\proof{Proof} $\bullet$ ${\rm \ol d)}$ is already known, and so is a) when 
$k=0$ (Th.~VI-13.3).
\medskip
\noindent$\bullet$ b) and ${\rm \ol b)}$ follow from Th.~10.6, and
c), ${\rm \ol c)}$ are equivalent to b), ${\rm \ol b)}$ via Serre duality:
$$H^{p,q}\big(P(V),\cO(k)\big)^\star=H^{n-p,n-q}\big(P(V),\cO(-k)\big),$$
thanks to the canonical isomorphism $\big(Z^{p,k}(V)\big)^\star=Z^{p,k}
(V^\star)$.
\medskip
\noindent$\bullet$ Let us prove now property a) when $k\ne 0$ and
property d). By Serre duality, we may assume $k>0$. Then
$$\Lambda^pT^\star P(V)\simeq K_{P(V)}\otimes\Lambda^{n-p}TP(V).$$
It is very easy to verify that $E\ge_\Nak0$ implies $\Lambda^sE
\ge_\Nak0$ for every integer $s$. Since $TP(V)\ge_\Nak0$,
we get therefore
$$F=\Lambda^{n-p}TP(V)\otimes\cO(k)>_\Nak0~~~~\hbox{\rm for}~~k>0,$$
and the Nakano vanishing theorem implies
$$\eqalignno{
H^{p,q}\big(P(V),\cO(k)\big)&=H^q\big(P(V),\Lambda^pT^\star P(V)\otimes
\cO(k)\big)\cr
&=H^q\big(P(V),K_{P(V)}\otimes F\big)=0,~~~~q\ge 1.&\square\cr}$$
\endproof

\section{11}{Ample Vector Bundles}
\subsection{11.A}{Globally Generated Vector Bundles}
All definitions concerning ampleness are purely algebraic  and do not 
involve differential geometry. We shall see however that ampleness is
intimately connected with the differential geometric notion of
positivity. For a general discussion of properties of ample vector
bundles in arbitrary characteristic,
we refer to \cite{Hartshorne 1966}.

\claim{(11.1) Definition} Let $E\to X$ be a holomorphic vector bundle over
an arbitrary complex manifold~$X$.
\smallskip
\item{\rm a)} $E$ is said to be globally generated if for 
every $x\in X$ the evaluation map $H^0(X,E) \to E_x$ is onto.
\smallskip
\item{\rm b)} $E$ is said to be semi-ample if there exists an
integer $k_0$ such that $S^kE$ is globally generated for $k\ge k_0$.
\smallskip
\endclaim

Any quotient of a trivial vector bundle is globally generated, for example
the tautological quotient vector bundle $Q$ over the Grassmannian $G_r(V)$
is globally generated. Conversely, every globally generated vector bundle
$E$ of rank $r$ is isomorphic to the quotient of a trivial vector bundle of
rank $\le n+r$, as shown by the following result.

\claim{(11.2) Proposition} If a vector bundle $E$ of rank $r$ is globally
generated, then there exists a finite dimensional subspace $V\subset
H^0(X,E)$, $\dim V\le n+r$, such that $V$ generates all fibers $E_x$, 
$x\in X$.
\endclaim

\proof{Proof} Put an arbitrary hermitian metric on $E$ and consider the 
Fr\'echet space
$\cF=H^0(X,E)^{\oplus n+r}$ of $(n+r)$-tuples of holomorphic sections
of $E$, endowed with the topology of uniform convergence on compact subsets
of~$X$. For every compact set $K\subset X$, we set
$$A(K)=
\{(s_1,\ldots,s_{n+r})\in\cF~{\rm which~do~not~generate~}E~{\rm on}~K\}.$$
It is enough to prove that $A(K)$ is of first category in $\cF\,:$
indeed, Baire's theorem will imply that $A(X)=
\bigcup A(K_\nu)$ is also of first category, if $(K_\nu)$ is an exhaustive
sequence of compact subsets of~$X$. It is clear that $A(K)$ is closed,
because $A(K)$ is characterized by the closed condition
$$\min_K\sum_{i_1<\cdots<i_r}|s_{i_1}\wedge\cdots\wedge s_{i_r}|=0.$$
It is therefore sufficient to prove that $A(K)$ has no interior point. By
hypothesis, each fiber $E_x$, $x\in K$, is generated by $r$ global
sections $s'_1,\ldots,s'_r$. We have in fact $s'_1\wedge\cdots\wedge s'_r\ne 0$
in a neighborhood $U_x$ of~$x$. By compactness of~$K$, there exist finitely
many sections $s'_1,\ldots,s'_N$ which generate $E$ in a neighborhood $\Omega$ 
of the set $K$.

If $T$ is a complex vector space of dimension $r$, define $R_k(T^p)$ as the set
of \hbox{$p$-tuples} $(x_1,\ldots,x_p)\in T^p$ of rank $k$. Given $a\in R_k(T^p)$,
we can reorder the $p$-tuple in such a way that $a_1\wedge\cdots\wedge a_k
\ne 0$. Complete these $k$ vectors
into a basis $(a_1,\ldots,a_k,b_1,\ldots,b_{r-k})$ of~$T$. For every point 
$x\in T^p$ in a neighborhood of~$a$, then $(x_1,\ldots,x_k,b_1,\ldots,b_{r-k})$ is
again a basis of~$T$. Therefore, we will have $x\in R_k(T^p)$ if and only if
the coordinates of $x_l$, $k+1\le l\le N$, relative to $b_1,\ldots,b_{r-k}$
vanish. It follows that $R_k(T^p)$ is a (non closed) submanifold of $T^p$ of 
codimension $(r-k)(p-k)$.

Now, we have a surjective affine bundle-morphism
$$\eqalign{\Phi:~\Omega\times\bC^{N(n+r)}&\longrightarrow E^{n+r}\cr
(x,\lambda)&\longmapsto\big(s_j(x)+\sum_{1\le k\le N}\lambda_{jk}s'_k(x)
\big)_{1\le j\le n+r}.\cr}$$
Therefore $\Phi^{-1}(R_k(E^{n+r}))$ is a locally trivial differentiable bundle
over $\Omega$, and the codimension of its fibers in $\bC^{N(n+r)}$ is 
$(r-k)(n+r-k)\ge n+1$ if $k<r\,;$ it follows
that the dimension of the total space $\Phi^{-1}(R_k(E^{n+r}))$ is
$\le N(n+r)-1$. By Sard's theorem 
$$\bigcup_{k<r}\cP_2\Big(\Phi^{-1}\big(R_k(E^{n+r})\big)\Big)$$
is of zero measure in $\bC^{N(n+r)}$. This means that for almost every value of
the parameter $\lambda$ the vectors $s_j(x)+\sum_k\lambda_{jk}s'_k(x)\in E_x$,
$1\le j\le n+r$, are of maximum rank $r$ at each point $x\in\Omega$. 
Therefore $A(K)$ has no interior point.\qed
\endproof

Assume now that $V\subset H^0(X,E)$ generates $E$ on $X$. Then there is an 
exact sequence
$$0\longrightarrow S\longrightarrow\soul V\longrightarrow E\longrightarrow 0\leqno(11.3)$$
of vector bundles over $X$, where $S_x=\{s\in V~;~s(x)=0\}$, 
$\codim_VS_x=r$. One obtains therefore a commutative diagram
$$\matrix{
E&\buildo{\displaystyle\Psi_V}\over\longrightarrow&Q\cr
\downarrow&&\downarrow\cr
X&\buildo{\displaystyle\psi_V}\over\longrightarrow&G_r(V)\cr}\leqno(11.4)$$
where $\psi_V,~\Psi_V$ are the holomorphic maps defined by
$$\eqalign{\psi_V(x)&=S_x,~~x\in X,\cr
           \Psi_V(u)&=\{s\in V~;~s(x)=u\}\in V/S_x,~~u\in E_x.}$$
In particular, we see that every globally generated vector bundle $E$ of
rank $r$ is the pull-back of the tautological quotient vector bundle $Q$
of rank $r$ over the Grassmannian by means of some holomorphic map
$X\longrightarrow G_r(V)$. In the special case when $\rk E=r=1$, the above diagram 
becomes
$$\matrix{
E&\buildo{\displaystyle\Psi_V}\over\longrightarrow&\cO(1)\cr
\downarrow&&\downarrow\cr
X&\buildo{\displaystyle\psi_V}\over\longrightarrow&P(V^\star)\cr}\leqno(11.4')$$

\claim{(11.5) Corollary} If $E$ is globally generated, then $E$ possesses a
hermitian metric such that $E\ge_\Grif0$ $($and also
$E^\star\le_\Nak 0)$.
\endclaim

\proof{Proof} Apply Prop.~6.11 to the exact sequence (11.3), where
$\soul V$ is endowed with an arbitrary hermitian metric.\qed
\endproof
 
When $E$ is of rank $r=1$, then $S^kE=E^{\otimes k}$ and any hermitian metric
of $E^{\otimes k}$ yields a metric on $E$ after extracting $k$-th roots.
Thus:

\claim{(11.6) Corollary} If $E$ is a semi-ample line bundle, then
$E\ge 0$.\qed
\endclaim

In the case of vector bundles ($r\ge 2$) the answer is unknown, mainly because
there is no known procedure to get a Griffiths semipositive metric on $E$ from 
one on $S^kE$. 

\subsection{11.B}{Ampleness}
We are now turning ourselves to the definition of ampleness. If $E\longrightarrow X$ is a 
holomorphic 
vector bundle, we define the bundle $J^kE$ of {\it $k$-jets of sections} of 
$E$ by $(J^kE)_x=\cO_x(E)/\big(\cM_x^{k+1}\cdot\cO_x(E)\big)$ for every 
$x\in X$, where $\cM_x$ is the maximal ideal of $\cO_x$. Let 
$(e_1,\ldots,e_r)$ be a holomorphic frame of $E$ and $(z_1,\ldots,z_n)$ analytic 
coordinates on an open subset $\Omega\subset X$. The fiber $(J^kE)_x$ can be 
identified with the set of Taylor developments of order $k\,:$
$$\sum_{1\le\lambda\le r,|\alpha|\le k}
c_{\lambda,\alpha}(z-x)^\alpha\,e_\lambda(z),$$
and the coefficients $c_{\lambda,\alpha}$ define coordinates along the
fibers of $J^kE$. It is clear that the choice of another holomorphic 
frame $(e_\lambda)$ would yield a linear change of coordinates 
$(c_{\lambda,\alpha})$ with holomorphic coefficients in $x$. Hence
$J^k E$ is a holomorphic vector bundle of rank $r{n+k\choose n}$.

\claim{(11.7) Definition} \smallskip
\item{\rm a)} $E$ is said to be very ample if all 
evaluation maps 
$$
H^0(X,E)\to (J^1E)_x,\quad H^0(X,E)\to E_x\oplus E_y,\quad
x,y\in X,~x\ne y,
$$
are surjective.
\smallskip
\item{\rm b)} $E$ is said to be ample if there exists an
integer $k_0$ such that $S^kE$ is very ample for $k\ge k_0$.\smallskip
\endclaim

\claim{(11.8) Example} \rm $\cO(1)\to\bP^n$ is a very ample line
bundle (immediate verification). Since the pull-back of a (very) ample
vector bundle by an embedding is clearly also (very) ample, diagram (V-16.8)
shows that $\Lambda^rQ\to G_r(V)$ is very ample. However, $Q$ itself cannot
be very ample if $r\ge 2$, because $\dim H^0(G_r(V),Q)=\dim V=d$, whereas
$${\rm rank}(J^1Q)=({\rm rank}\,Q)\big(1+\dim G_r(V)\big)=r\big(1+r(d-r)\big)>d
~~{\rm if}~~r\ge 2.$$
\endclaim

\claim{(11.9) Proposition} If $E$ is very ample of rank $r$, there
exists a subspace $V$ of $H^0(X,E)$, $\dim V\le\max\big(nr+n+r,2(n+r)\big)$,
such that all the evaluation maps $V\to E_x\oplus E_y$, $x\ne y$, and 
$V\to (J^1E)_x$, $x\in X$, are surjective.
\endclaim

\proof{Proof} The arguments are exactly the same as in the proof of
Prop.~11.4, if we consider instead the bundles $J^1E\longrightarrow X$ and
$E\times E\longrightarrow X\times X\setminus\Delta_X$ of respective ranks $r(n+1)$ and
$2r$, and sections $s'_1,\ldots,s'_N\in H^0(X,E)$ generating these 
bundles.\qed
\endproof

\claim{(11.10) Proposition} Let $E\to X$ be a holomorphic vector bundle.
\smallskip
\item{\rm a)} If $V\subset H^0(X,E)$ generates $J^1E\longrightarrow X$ and
$E\times E\longrightarrow X\times X\setminus\Delta_X$, then $\psi_V$ is an embedding.
\smallskip
\item{\rm b)} Conversely, if rank$\,E=1$ and if there exists
$V\subset H^0(X,E)$ generating $E$ such that $\psi_V$ is an embedding, then
$E$ is very ample.\smallskip
\endclaim

\proof{Proof} b) is immediate, because $E=\psi_V^\star(\cO(1))$ and
$\cO(1)$ is very ample. Note that the result is false for $r\ge 2$ as shown
by the example $E=Q$ over $X=G_r(V)$.
\medskip
\noindent a) Under the assumption of a), it is clear since $S_x=\{s\in V~;~
s(x)=0\}$ that $S_x=S_y$ implies $x=y$, hence $\psi_V$ is injective. Therefore,
it is enough to prove that the map $x\mapsto S_x$ has an injective differential.
Let $x\in X$ and $W\subset V$ such that $S_x\oplus W=V$. Choose
a coordinate system in a neighborhood of $x$ in $X$ and a small tangent vector 
$h\in T_xX$. The element $S_{x+h}\in G_r(V)$ is the graph of a small
linear map $u=O(|h|):S_x\to W$. Thus we have
$$S_{x+h}=\{s'=s+t\in V~;~s\in S_x,~t=u(s)\in W,~s'(x+h)=0\}.$$
Since $s(x)=0$ and $|t|=O(|h|)$, we find 
$$s'(x+h)=s'(x)+d_xs'\cdot h+O(|s'|\cdot|h|^2)=t(x)+d_xs\cdot h+
O(|s|\cdot|h|^2),$$
thus $s'(x+h)=0$ if and only if $t(x)=-d_xs\cdot h+O(|s|\cdot|h|^2)$.
Thanks to the fiber isomorphism 
$\Psi_V:E_x\to V/S_x\simeq W$,~ $t(x)\longmapsto t$ mod $S_x$, we get:
$$u(s)=t=\Psi_V(t(x))=-\Psi_V\big(d_xs\cdot h+O(|s|\cdot|h|^2)\big).$$
Recall that $T_yG_r(V)=\hom(S_y,Q_y)=\hom(y,V/y)$ (see V-16.5) and use these
identifications at $y=S_x$. It follows that
$$(d_x\psi_V)\cdot h=u=\big(S_x\to V/S_x,~s\longmapsto -\Psi_V
(d_xs\cdot h)\big),\leqno(11.11)$$
Now hypothesis a) implies that $S_x\ni s\longmapsto
d_xs\in\hom(T_xX,E_x)$ is onto, hence $d_x\psi_V$ is injective.\qed
\endproof

\claim{(11.12) Corollary} If $E$ is an ample line bundle, then $E>0$.
\endclaim

\proof{Proof} If $E$ is very ample, diagram $(11.4')$ shows that $E$ is the
pull-back of $\cO(1)$ by the embedding $\psi_V$, hence $i\Theta(E)=
\psi_V^\star\big(\ii\Theta(\cO(1))\big)>0$ with the induced metric.
The ample case follows by extracting roots.\qed
\endproof

\claim{(11.13) Corollary} If $E$ is a very ample vector bundle, then $E$
carries a hermitian metric such that $E^\star<_\Nak 0$,
in particular $E>_\Grif 0$.
\endclaim

\proof{Proof} Choose $V$ as in Prop.~11.9 and select an arbitrary
hermitian metric on $V$. Then diagram 11.4 yields $E=\psi_V^\star Q$,
hence $\theta_E=\Psi_V^\star\theta_Q$. By formula (V-16.9) we have
for every $\xi\in TG_r(V)=\hom(S,Q)$ and $t\in Q\,:$
$$\theta_Q(\xi\otimes t,\xi\otimes t)=\sum_{j,k,l}
\xi_{jk}\ol\xi_{lk}t_l\ol t_j
=\sum_k\Big|\sum_j\ol t_j\xi_{jk}\Big|^2=|\langle\bu,t\rangle\circ\xi|^2.$$
Let $h\in T_xX$, $t\in E_x$. Thanks to formula (11.11), we get
$$\eqalign{
\theta_E(h\otimes t&,h\otimes t)=\theta_Q\big((d_x\psi_V\cdot h)\otimes
\Psi_V(t),(d_x\psi_V\cdot h)\otimes\Psi_V(t)\big)\cr
&=\big|\langle\bu,\Psi_V(t)\rangle\circ(d_x\psi_V\cdot h)\big|^2
=\big|S_x\ni s\longmapsto\langle\Psi_V(d_xs\cdot h),\Psi_V(t)\rangle\big|^2\cr
&=\big|S_x\ni s\longmapsto\langle d_xs\cdot h,t\rangle\big|^2\ge 0.\cr}$$
As $S_x\ni s\mapsto d_xs\in T^\star X\otimes E$ is surjective, it follows 
that \hbox{$\theta_E(h\otimes t,h\otimes t)\ne 0$} when $h\ne 0$,
$t\ne 0$. Now, $d_xs$ defines a linear form on $TX\otimes E^\star$ and the
above formula for the curvature of $E$ clearly yields
$$\theta_{E^\star}(u,u)=-|S_x\ni s\longmapsto d_xs\cdot u|^2<0~~~~{\rm if}~
u\ne 0.\eqno{\square}$$
\endproof

\claim{(11.14) Problem (\cite{Griffiths 1969})} If $E$ is an ample
vector bundle over a compact manifold~$X$, then is $E>_\Grif 0$~?
\endclaim

Griffiths' problem has been solved in the affirmative when $X$ is a
curve (see \cite{Ume\-mura~1973} and also \cite{Campana-Flenner~1990}),
but the general case is still unclear and seems very deep. The next
sections will be concerned with the important result of Kodaira
asserting the equivalence between positivity and ampleness for line
bundles.

\section{12}{Blowing-up along a Submanifold}
Here we generalize the blowing-up process already considered in Remark 4.5
to arbitrary manifolds. Let $X$ be a complex $n$-dimensional manifold and
$Y$ a closed submanifold with $\codim_XY=s$.

\claim{(12.1) Notations} The normal bundle of $Y$ in $X$ is the vector
bundle over $Y$ defined as the quotient $NY=(TX)_{\restriction Y}/TY$.
The fibers of $NY$ are thus given by $N_yY=T_yX/T_yY$ at every point $y\in Y$.
We also consider the projectivized normal bundle
$P(NY)\to Y$ whose fibers are the projective spaces $P(N_yY)$ associated
to the fibers of $NY$.
\endclaim

The {\it blow-up of $X$ with center $Y$}
(to be constructed later) is a complex $n$-dimensional manifold $\wt X$ 
together with a holomorphic map $\sigma:\wt X\longrightarrow X$ such that:
\smallskip
\item{i)} $E:=\sigma^{-1}(Y)$ is a smooth {\it hypersurface} in 
$\wt X$, and the restriction \hbox{$\sigma:E\to Y$}
is a holomorphic fiber bundle isomorphic to the
projec\-ti\-vized normal bundle \hbox{$P(NY)\to Y$}.
\smallskip
\item{ii)} $\sigma:\wt X\setminus E\longrightarrow X\setminus Y$ is a biholomorphism.
\smallskip
\noindent
In order to construct $\wt X$ and $\sigma$, we first define the set-theoretic
underlying objects as the disjoint sums
$$\cmalign{
\wt X&=(X\setminus Y)\amalg E,~~~~
&{\rm where}~~E:=P(NY),\cr
\sigma&=\Id_{X\setminus Y}\amalg{}\,\,\pi,~~~~
&{\rm where}~~\pi:E\longrightarrow Y.\cr}$$

\vbox{%
\input epsfiles/fig_7_1.tex
\vskip16mm
\centerline{{\bf VII-1} Blow-up of one point in $X$.}}
\vskip5mm

\noindent
This means intuitively that we have replaced each point $y\in Y$ by the
projective space of all directions normal to $Y$. When $Y$ is reduced to
a single point, the geometric picture is given by Fig.~1 below.
In general, the picture is obtained by slicing $X$ transversally to $Y$
near each point and by blowing-up each slice at the intersection point
with~$Y$.

It remains to construct the manifold structure on $\wt X$ and in particular
to describe what are the holomorphic functions near a point of~$E$.
Let $f,g$ be holomorphic functions on an open set $U\subset X$ such that 
\hbox{$f=g=0$} on $Y\cap U$. Then $df$ and $dg$ vanish on $TY_{\restriction 
Y\cap U}$, hence $df$ and $dg$ induce linear forms on $NY_{\restriction 
Y\cap U}$. The holomorphic function $h(z)=f(z)/g(z)$ on the open set
$$U_g:=\big\{z\in U~;~g(z)\ne 0\big\}\subset U\setminus Y$$
can be extended in a natural way to a function $\wt h$ on the set
$$\wt U_g=U_g\cup\big\{(z,[\xi])\in P(NY)_{\restriction Y\cap U}~;~dg_z(\xi)
\ne 0\big\}\subset\wt X$$
by letting
$$\wt h(z,[\xi])={df_z(\xi)\over dg_z(\xi)},~~~~(z,[\xi])\in P(NY)_{\restriction
Y\cap U}.$$
Using this observation, we now define the manifold structure on $\wt X$ by
giving explicitly an atlas. Every coordinate chart of $X\setminus Y$ is taken 
to be also a coordinate chart of $\wt X$. Furthermore, for every point
$y_0\in Y$, there exists a neighborhood $U$ of $y_0$ in $X$ and a
coordinate chart $\tau(z)=(z_1,\ldots,z_n):U\to\bC^n$ centered at $y_0$ such that
$\tau(U)=B'\times B''$ for some balls \hbox{$B'\subset\bC^s$},
$B''\subset\bC^{n-s}$, and such that 
\hbox{$Y\cap U=\tau^{-1}(\{0\}\times B'')=\{z_1{=}\ldots{=}z_s{=}0\}$.}
It~follows that $(z_{s+1},\ldots,z_n)$ are local coordinates on $Y\cap U$
and that the vector fields
$(\partial/\partial z_1,\ldots,\partial/\partial z_s)$ yield a holomorphic frame
of $NY_{\restriction Y\cap U}$. Let us denote by $(\xi_1,\ldots,\xi_s)$ the
corresponding coordinates along the fibers of $NY$. Then
$(\xi_1,\ldots,\xi_s,z_{s+1},\ldots,z_n)$ are coordinates on the total space
$NY$. For every $j=1,\ldots,s$, we set
$$\wt U_j=\wt U_{z_j}=\big\{z\in U\setminus Y~;~z_j\ne 0\big\}\cup
\big\{(z,[\xi])\in P(NY)_{\restriction Y\cap U}~;~\xi_j\ne 0\big\}.$$
Then $(\wt U_j)_{1\le j\le s}$ is a covering of $\wt U=\sigma^{-1}(U)$
and for each $j$ we define a coordinate chart $\wt\tau_j=(w_1,\ldots,w_n):
\wt U_j\longrightarrow\bC^n$ by
$$w_k:=\Big({z_k\over z_j}\Big)^\sim~~~~{\rm for}~~1\le k\le s,k\ne j\,;~~~~
w_k:=z_k~~~{\rm for}~~k>s~~{\rm or}~~k=j.$$
For $z\in U\setminus Y$, resp.\ $(z,[\xi])\in P(NY)_{\restriction Y\cap U}$,
we get
$$\eqalign{
\wt\tau_j(z)&=(w_1,\ldots,w_n)=\Big({z_1\over z_j},\ldots,{z_{j-1}\over z_j},z_j,
{z_{j+1}\over z_j},\ldots,{z_s\over z_j},z_{s+1},\ldots,z_n\Big),\cr
\wt\tau_j(z,[\xi])&=(w_1,\ldots,w_n)=\Big({\xi_1\over \xi_j},\ldots,{\xi_{j-1}\over
\xi_j},\,0\,,{\xi_{j+1}\over\xi_j}
,\ldots,{\xi_s\over \xi_j},\xi_{s+1},\ldots,\xi_n\Big).\cr}$$
With respect to the coordinates $(w_k)$ on $\wt U_j$ and $(z_k)$ on $U$,
the map $\sigma$ is given by
$$\leqalignno{
\wt U_j&\buildo\sigma\over\longrightarrow U&(12.2)\cr
w&\buildo\sigma_j\over\longmapsto
(w_1w_j,\ldots,w_{j-1}w_j\,;\,w_j\,;\,w_{j+1}w_j,\ldots,w_sw_j\,;\,w_{s+1},\ldots,w_n)\cr}$$
where $\sigma_j=\tau\circ\sigma\circ\wt\tau_j^{-1}$, thus
$\sigma$ is holomorphic. The range of the coordinate chart
$\wt\tau_j$ is $\wt\tau_j(\wt U_j)=\sigma_j^{-1}\big(\tau(U)\big)$, so it is
actually open in $\bC^n$. Furthermore $E\cap\wt U_j$ is defined by the
single equation $w_j=0$, thus $E$ is a smooth hypersurface in $\wt X$.
It remains only to verify that the coordinate changes $w\longmapsto w'$
associated to any coordinate change $z\longmapsto z'$ on $X$ are
holomorphic. For that purpose, it is sufficient to verify that
$(f/g)^\sim$ is holomorphic in $(w_1,\ldots,w_n)$ on $\wt U_j\cap\wt U_g$.
As $g$ vanishes on $Y\cap U$, we can write $g(z)=\sum_{1\le k\le s}
z_kA_k(z)$ for some holomorphic functions $A_k$ on $U$. Therefore
$${g(z)\over z_j}=A_j(\sigma_j(w))+\sum_{k\ne j}w_kA_k(\sigma_j(w))$$
has an extension $(g/z_j)^\sim$ to $\wt U_j$ which is a holomorphic function
of the variables\break $(w_1,\ldots,w_n)$. Since $(g/z_j)^\sim(z,[\xi])=
dg_z(\xi)/\xi_j$ on $E\cap\wt U_j$, it is clear that
$$\wt U_j\cap\wt U_g=\big\{w\in\wt U_j~;~(g/z_j)^\sim(w)\ne 0\big\}.$$
Hence $\wt U_j\cap\wt U_g$ is open in $\wt U_g$ and $(f/g)^\sim=
(f/z_j)^\sim/(g/z_j)^\sim$ is holomorphic on $\wt U_j\cap\wt U_g$.

\claim{(12.3) Definition} The map $\sigma:\wt X\to X$ is called the blow-up
of $X$ with center $Y$ and $E=\sigma^{-1}(Y)\simeq P(NY)$ is called the
exceptional divisor of $\wt X$.
\endclaim

According to (V-13.5), we denote by $\cO(E)$ the line bundle on $\wt X$
associated to the divisor $E$ and by $h\in H^0(\wt X,\cO(E))$
the canonical section such that ${\rm div}(h)=[E]$.
On the other hand, we denote by $\cO_{P(NY)}(-1)\subset\pi^\star(NY)$ the
tautological line subbundle over $E=P(NY)$ such that the fiber above the
point $(z,[\xi])$ is $\bC\xi\subset N_zY$.

\claim{(12.4) Proposition} $\cO(E)$ enjoys the following properties:
\smallskip
\item{\rm a)} $\cO(E)_{\restriction E}$ is isomorphic to $\cO_{P(NY)}(-1)$.
\smallskip
\item{\rm b)} Assume that $X$ is compact. For every positive line bundle 
$L$ over $X$, the line bundle $\cO(-E)\otimes\sigma^\star(L^k)$ over $\wt X$
is positive for $k>0$ large enough.\smallskip
\endclaim

\proof{Proof} a) The canonical section $h\in H^0(\wt X,\cO(E))$ vanishes at order
$1$ along~$E$, hence the kernel of its differential
$$dh:(T\wt X)_{\restriction E}\longrightarrow\cO(E)_{\restriction E}$$
is $TE$. We get therefore an isomorphism $NE\simeq \cO(E)_{\restriction E}$.
Now, the map \hbox{$\sigma:\wt X\to X$} satisfies $\sigma(E)\subset Y$,
so its differential $d\sigma:T\wt X\longrightarrow\sigma^\star(TX)$ is such that
$d\sigma(TE)\subset\sigma^\star(TY)$. Therefore $d\sigma$ induces a morphism
$$NE\longrightarrow\sigma^\star(NY)=\pi^\star(NY)\leqno(12.5)$$
of vector bundles over $E$. The vector field $\partial/
\partial w_j$ yields a non vanishing section of $NE$ on $\wt U_j$,
and $(12.2)$ implies
$$d\sigma_j\Big({\partial\over\partial w_j}\Big)=
{\partial\over\partial z_j}+\sum_{1\le k\le s,k\ne j}w_k
{\partial\over\partial z_k}~~~~/\!/~~~
\sum_{1\le k\le s}\xi_k{\partial\over\partial z_k}$$
at every point $(z,[\xi])\in E$. This shows that (12.5) is an
isomorphism of $NE$ onto $\cO_{P(NY)}(-1)\subset\pi^\star(NY)$, hence
$$\cO(E)_{\restriction E}\simeq NE\simeq\cO_{P(NY)}(-1).\leqno(12.6)$$
{\rm b)} Select an arbitrary hermitian metric on $TX$ and
consider the induced metrics on $NY$ and on $\cO_{P(NY)}(1)\longrightarrow E=P(NY)$. 
The restriction of $\cO_{P(NY)}(1)$ to each fiber $P(N_zY)$ is the standard line
bundle $\cO(1)$ over $\bP^{s-1}\,;$ thus by (V-15.10) this restriction has a 
positive definite curvature form. Extend now the metric of
$\cO_{P(NY)}(1)$ on $E$ to a metric of $\cO(-E)$ on $X$ in an arbitrary way.
If $F=\cO(-E)\otimes\sigma^\star(L^k)$, then $\Theta(F)=\Theta(\cO(-E))+
k\,\sigma^\star \Theta(L)$, thus for every $t\in T\wt X$ we have
$$\theta_F(t,t)=\theta_{\cO(-E)}(t,t)+
k\,\theta_L\big(d\sigma(t),d\sigma(t)\big).$$
By the compactness of the unitary tangent bundle to $\wt X$ and the positivity
of $\theta_L$, it is sufficient to verify that $\theta_{\cO(-E)}(t,t)>0$ for
every unit vector $t\in T_z\wt X$ such that $d\sigma(t)=0$. However, from
the computations of a), this can only happen when $z\in E$ and 
$t\in TE$, and in that case $d\sigma(t)=d\pi(t)=0$, so $t$
is tangent to the fiber $P(N_zY)$. Therefore
$$\theta_{\cO(-E)}(t,t)=\theta_{\cO_{P(NY)}(1)}(t,t)>0.\eqno\square$$
\endproof

\claim{(12.7) Proposition} The canonical line bundle of $\wt X$ is
given by
$$K_{\wt X}=\cO\big((s-1)E\big)\otimes\sigma^\star K_X,~~~~\hbox{\it where}~~
s=\codim_XY.$$
\endclaim

\proof{Proof} $K_X$ is generated on $U$ by the holomorphic $n$-form
$dz_1\wedge\ldots\wedge dz_n$. Using (12.2), we see that
$\sigma^\star K_X$ is generated on $\wt U_j$ by
$$\sigma^\star(dz_1\wedge\ldots\wedge dz_n)=w_j^{s-1}\,dw_1\wedge\ldots
\wedge dw_n.$$
Since the divisor of the section $h\in H^0(\wt X,\cO(E))$ is the
hypersurface $E$ defined by the equation $w_j=0$ in $\wt U_j$,
we have a well defined line bundle isomorphism
$$\sigma^\star K_X\longrightarrow \cO\big((1-s)E\big)\otimes K_{\wt X},~~~~~
\alpha\longmapsto h^{1-s}\sigma^\star(\alpha).\eqno\square$$
\endproof

\section{13}{Equivalence of Positivity and Ampleness for Line Bundles}
We have seen in section 11 that every ample line bundle carries a hermitian
metric of positive curvature. The converse will be a consequence of the
following result.

\claim{(13.1) Theorem} Let $L\longrightarrow X$ be a positive line bundle and $L^k$ the 
\hbox{$k$-th} tensor power of $L$. For every
$N$-tuple $(x_1,\ldots,x_N)$ of distinct points of $X$, there exists a
constant $C>0$ such that the evaluation maps
$$H^0(X,L^k)\longrightarrow(J^mL^k)_{x_1}\oplus\cdots\oplus(J^mL^k)_{x_N}$$
are surjective for all integers $m\ge 0$, $k\ge C(m+1)$.
\endclaim

\claim{(13.2) Lemma} Let $\sigma:\wt X\longrightarrow X$ be the blow-up of $X$ with
center the finite set $Y=\{x_1,\ldots,x_N\}$, and let $\cO(E)$ be the
line bundle associated to the exceptional divisor $E$. Then
$$H^1(\wt X,\cO(-mE)\otimes\sigma^\star L^k)=0$$
for $m\ge 1$, $k\ge Cm$ and $C\ge 0$ large enough.
\endclaim

\proof{Proof} By Prop.~12.7 we get $K_{\wt X}=\cO\big((n-1)E\big)\otimes
\sigma^\star K_X$ and
$$H^1\big(\wt X,\cO(-mE)\otimes\sigma^\star L^k\big)
=H^{n,1}\big(\wt X,K_{\wt X}^{-1}\otimes\cO(-mE)\otimes\sigma^\star L^k\big)
=H^{n,1}\big(\wt X,F\big)$$
where $F=\cO\big(-(m+n-1)E\big)\otimes\sigma^\star(K_X^{-1}\otimes L^k)$,
so the conclusion will follow from the Kodaira-Nakano vanishing theorem if
we can show that $F>0$ when $k$ is large enough. Fix an arbitrary hermitian
metric on $K_X$. Then
$$\Theta(F)=(m+n-1)\Theta(\cO(-E))+
\sigma^\star\big(k\Theta(L)-\Theta(K_X)\big).$$
There is $k_0\ge 0$ such that $\ii\big(k_0\Theta(L)-\Theta(K_X)\big)>0$
on~$X$, and Prop.~12.4 implies the existence of $C_0>0$ such that
$\ii\big(\Theta(\cO(-E))+C_0\sigma^\star \Theta(L)\big)>0$ on~$\wt X$. Thus
$\ii\Theta(F)>0$ for $m\ge 2-n$ and $k\ge k_0+C_0(m+n-1)$.\qed
\endproof

\proof{Proof of Theorem 13.1.} Let $v_j\in H^0(\Omega_j,L^k)$ be a 
holomorphic section of $L^k$ in a neighborhood $\Omega_j$ of $x_j$ having a 
prescribed $m$-jet at $x_j$. Set
$$v(x)=\sum_j\psi_j(x)v_j(x)$$
where $\psi_j=1$ in a neighborhood of $x_j$ and $\psi_j$ has compact support
in $\Omega_j$. Then $d''v=\sum d''\psi_j\cdot v_j$ vanishes in a neighborhood
of $x_1,\ldots,x_N$. Let $h$ be the canonical section of $\cO(E)^{-1}$ such that
${\rm div}(h)=[E]$. The $(0,1)$-form $\sigma^\star d''v$ vanishes in a
neighborhood of $E=h^{-1}(0)$, hence
$$w=h^{-(m+1)}\sigma^\star d''v\in \cC^\infty_{0,1}\big(\wt X,\cO(-(m+1)E)\otimes
\sigma^\star L^k\big).$$
and $w$ is a $d''$-closed form. By Lemma~13.2 there exists a smooth section
$$
u\in \cC^\infty_{0,0}\big(\wt X,\cO(-(m+1)E)\otimes\sigma^\star L^k\big)
$$
such that $d''u=w=h^{-(m+1)}\sigma^\star d''v$. This implies
$$\sigma^\star v-h^{m+1}u\in H^0(\wt X,\sigma^\star L^k),$$
and since $\sigma^\star L$ is trivial near $E$, there exists a section
$g\in H^0(X,L^k)$ such that $\sigma^\star g=\sigma^\star v-h^{m+1}u$.
As $h$ vanishes at order 1 along $E$, the $m$-jet of $g$
at $x_j$ must be equal to that of $v$ (or $v_j$).\qed
\endproof

\claim{(13.3) Corollary} For any holomorphic line bundle $L\longrightarrow X$, the
following conditions are equivalent:
\smallskip
\item{\rm a)} $L$ is ample;
\smallskip
\item{\rm b)} $L>0$, i.e.\ $L$ possesses a hermitian metric such that
$\ii\Theta(L)>0$.
\endclaim

\proof{Proof} a) $\Longrightarrow$ b) is given by Cor.~11.12,
whereas b) $\Longrightarrow$ a) is a consequence of Th.~13.1 for
$m=1$.\qed
\endproof

\section{14}{Kodaira's Projectivity Criterion}
The following fundamental projectivity criterion is due to 
\cite{Kodaira 1954}.

\claim{(14.1) Theorem} Let $X$ be a compact complex manifold, $\dim_\bC X=n$.
The following conditions are equivalent.
\smallskip
\item{\rm a)} $X$ is projective algebraic, i.e.\ $X$ can be embedded as an
algebraic submanifold of the complex projective space $\bP^N$ for $N$ large.
\smallskip
\item{\rm b)} $X$ carries a positive line bundle $L$.
\smallskip
\item{\rm c)} $X$ carries a Hodge metric,
i.e.\ a K\"ahler metric $\omega$ with rational cohomology class
$\{\omega\}\in H^2(X,\bQ)$.\smallskip
\endclaim

\proof{Proof} a) $\Longrightarrow$ b). Take $L=\cO(1)_{\restriction X}$.
\medskip
\noindent b) $\Longrightarrow$ c). Take $\omega={\ii\over 2\pi}\Theta(L)\,;$
then $\{\omega\}$ is the image of $c_1(L)\in H^2(X,\bZ)$.
\medskip
\noindent c) $\Longrightarrow$ b). We can multiply $\{\omega\}$ by a common
denominator of its coefficients and suppose that $\{\omega\}$ is in the image
of $H^2(X,\bZ)$. Then Th.~V-13.9~b) shows that there exists a hermitian
line bundle $L$ such that ${\ii\over 2\pi}\Theta(L)=\omega >0$.
\medskip
\noindent b) $\Longrightarrow$ a). Corollary 13.3 shows that $F=L^k$ is very 
ample for some integer $k>0$. Then Prop.~11.9 enables us to find a subspace
$V$ of $H^0(X,F)$, $\dim V\le 2n+2$, such that
$\psi_V:X\longrightarrow G_1(V)=P(V^\star)$ is an embedding. Thus $X$ can
be embedded in $\bP^{2n+1}$ and Chow's theorem~II-7.10 shows that the
image is an algebraic set in~$\bP^{2n+1}$.\qed
\endproof

\claim{(14.2) Remark} \rm The above proof shows in particular that every
$n$-dimen\-sional projective manifold $X$ can be embedded in $\bP^{2n+1}$.
This can be shown directly by using generic projections
$\bP^N\to\bP^{2n+1}$ and Whitney type arguments as in~11.2.
\endclaim

\claim{(14.3) Corollary} Every compact Riemann surface $X$ is isomorphic to
an algebraic curve in $\bP^3$.
\endclaim

\proof{Proof} Any positive smooth form $\omega$ of type $(1,1)$ is
K\"ahler, and $\omega$ is in fact a Hodge metric if we normalize
its volume so that $\int_X\omega=1$.\qed
\endproof

\noindent This example can be somewhat generalized as follows.

\claim{(14.4) Corollary} Every K\"ahler manifold $(X,\omega)$ such that
$H^2(X,\cO)=0$ is projective.
\endclaim

\proof{Proof} By hypothesis $H^{0,2}(X,\bC)=0=H^{2,0}(X,\bC)$, hence
$$H^2(X,\bC)=H^{1,1}(X,\bC)$$
admits a basis
$\{\alpha_1\},\ldots,\{\alpha_N\}\in H^2(X,\bQ)$ where $\alpha_1,\ldots,\alpha_N$
are harmonic real $(1,1)$-forms. Since $\{\omega\}$ is real, we have
$\{\omega\}=\lambda_1\{\alpha_1\}+\ldots+\lambda_N\{\alpha_N\}$, 
$\lambda_j\in\bR$, thus
$$\omega=\lambda_1\alpha_1+\ldots+\lambda_N\alpha_N$$
because $\omega$ itself is harmonic. If $\mu_1,\ldots,\mu_N$ are
rational numbers sufficiently close to $\lambda_1,\ldots,\lambda_N$, then
$\wt\omega:=\mu_1\alpha_1+\cdots\mu_N\alpha_N$ is close to $\omega$,
so $\wt\omega$ is a positive definite $d$-closed $(1,1)$-form, and 
$\{\wt\omega\}\in H^2(X,\bQ)$.\qed
\endproof

We obtain now as a consequence the celebrated Riemann criterion characterizing
{\it abelian varieties} (${}={}$projective algebraic complex tori).

\claim{(14.5) Corollary} A complex torus $X=\bC^n/\Gamma$ $(\Gamma$ a lattice
of $\bC^n)$ is an abelian variety if and only if there exists a positive
definite hermitian form $h$ on $\bC^n$ such that
$$\Im\big(h(\gamma_1,\gamma_2)\big)\in\bZ~~~~{\rm for~all}~~\gamma_1,\gamma_2
\in\Gamma.$$
\endclaim

\proof{Proof (Sufficiency of the condition).} Set $\omega=-\Im h$. Then $\omega$
defines a constant K\"ahler metric on $\bC^n$, hence also on
$X=\bC^n/\Gamma$. Let $(a_1,\ldots,a_{2n})$ be an integral basis of the lattice
$\Gamma$. We denote by $T_j$, $T_{jk}$ the real $1$- and $2$-tori
$$T_{j}=(\bR/\bZ)a_j,~~~1\le j\le n,~~~~T_{jk}=T_j\oplus T_k,~~~
1\le j<k\le 2n.$$
Topologically we have $X\approx T_1\times\ldots\times T_{2n}$, so the
K\"unneth formula IV-15.7 yields
$$\eqalign{
&H^\bu(X,\bZ)\simeq\bigotimes_{1\le j\le 2n}\big(H^0(T_j,\bZ)\oplus 
H^1(T_j,\bZ)\big),\cr
&H^2(X,\bZ)\simeq\bigoplus_{1\le j<k\le 2n}H^1(T_j,\bZ)\otimes
H^1(T_k,\bZ)\simeq\bigoplus_{1\le j<k\le 2n}H^2(T_{jk},\bZ)\cr}$$
where the projection $H^2(X,\bZ)\longrightarrow H^2(T_{jk},\bZ)$ is induced by the 
injection $T_{jk}\subset X$. In the identification
$H^2(T_{jk},\bR)\simeq\bR$, we get
$$\{\omega\}_{\restriction\,T_{jk}}=\int_{T_{jk}}\omega=\omega(a_j,a_k)=
-\Im h(a_j,a_k).\leqno(14.6)$$
The assumption on $h$ implies $\{\omega\}_{\restriction\,T_{jk}}\in
H^2(T_{jk},\bZ)$ for all $j,k$, therefore
$\{\omega\}\in H^2(X,\bZ)$ and $X$ is projective by Th.~(14.1).
\endproof

\proof{Proof (Necessity of the condition).} If $X$ is projective, then $X$ admits
a K\"ahler metric $\omega$ such that $\{\omega\}$ is in the image of
$H^2(X,\bZ)$. In general, $\omega$ is not invariant under the
translations $\tau_x(y)=y-x$ of $X$. Therefore, we replace $\omega$ by 
its ``mean value'':
$$\wt\omega={1\over{\rm Vol}(X)}\int_{x\in X}(\tau_x^\star\omega)\,dx,$$
which has the same cohomology class as $\omega$ ($\tau_x$ is homotopic
to the identity). Now $\wt\omega$ is the imaginary part of a constant 
positive definite hermitian form $h$ on $\bC^n$, and formula
(14.6) shows that $\Im h(a_j,a_k)\in\bZ$.\qed
\endproof

\claim{(14.7) Example} {\rm Let $X$ be a projective manifold. We shall prove
that the Jacobian ${\rm Jac}(X)$ and the Albanese variety ${\rm Alb}(X)$ 
(cf.\ \S~VI-13 for definitions) are abelian varieties.

In fact, let $\omega$ be a K\"ahler  metric on $X$ such that $\{\omega\}$
is in the image of $H^2(X,\bZ)$ and let $h$ be the hermitian metric on
$H^1(X,\cO)\simeq H^{0,1}(X,\bC)$ defined by 
$$h(u,v)=\int_X -2\ii\,u\wedge\ol v\wedge\omega^{n-1}$$
for all closed $(0,1)$-forms $u,v$. As
$$-2\ii\,u\wedge\ol v\wedge\omega^{n-1}={2\over n}\,|u|^2\,\omega^n,$$
we see that $h$ is a positive definite hermitian form on $H^{0,1}(X,\bC)$.
Consider elements $\gamma_j\in H^1(X,\bZ)$, $j=1,2$. If we write
$\gamma_j=\gamma'_j+\gamma''_j$ in the decomposition 
$H^1(X,\bC)=H^{1,0}(X,\bC)\oplus H^{0,1}(X,\bC)$, we get
$$\eqalign{
h(\gamma''_1,\gamma''_2)&=\int_X -2\ii\,\gamma''_1\wedge\gamma'_2\wedge
\omega  ^{n-1},\cr
\Im h(\gamma''_1,\gamma''_2)&=\int_X (\gamma'_1\wedge\gamma''_2+
\gamma''_1\wedge\gamma'_2)\wedge\omega^{n-1}
=\int_X \gamma_1\wedge\gamma_2\wedge\omega^{n-1}\in\bZ.\cr}$$
Therefore ${\rm Jac}(X)$ is an abelian variety.

Now, we observe that $H^{n-1,n}(X,\bC)$ is the anti-dual of $H^{0,1}(X,\bC)$
by Serre duality. We select on $H^{n-1,n}(X,\bC)$ the dual hermitian metric
$h^\star$. Since the Poincar\'e bilinear pairing yields a unimodular
bilinear map
$$H^1(X,\bZ)\times H^{2n-1}(X,\bZ)\longrightarrow\bZ,$$
we easily conclude that $\Im h^\star(\gamma''_1,\gamma''_2)\in\bQ$ for
all $\gamma_1,\gamma_2\in H^{2n-1}(X,\bZ)$. Therefore ${\rm Alb}(X)$ is
also an abelian variety.}
\endclaim


\chapter{Chapter VIII}{$L^2$ Estimates on Pseudoconvex Manifolds}
\begineightpoint
The main goal of this chapter is to show that the differential
geometric technique that has been used in order to prove vanishing
theorems also yields very precise $L^2$ estimates for the solutions of
equations $d''u=v$ on pseudoconvex manifolds. The central idea, due to
\cite{H\"ormander~1965}, is to introduce weights of the type $e^{-\varphi}$
where $\varphi$ is a function satisfying suitable convexity conditions.
This method leads to generalizations of many standard vanishing
theorems to weakly pseudoconvex manifolds. As a special case, we obtain
the original H\"ormander estimates for pseudoconvex domains of
$\bC^n$, and give some applications to algebraic geometry
(H\"ormander-Bombieri-Skoda theorem, properties of zero sets of
polynomials in $\bC^n$).  We also derive the Ohsawa-Takegoshi
extension theorem for $L^2$ holomorphic functions and Skoda's $L^2$
estimates for surjective bundle morphisms \cite{Skoda 1972a, 1978},
\cite{Demailly~1982c}. Skoda's estimates can be used to obtain a quick
solution of the Levi problem, and have important applications
to local algebra and Nullstellensatz theorems. Finally, $L^2$ estimates
are used to prove the Newlander-Nirenberg theorem on the analyticity of
almost complex structures. We apply it to establish Kuranishi's theorem
on deformation theory of compact complex manifolds.\endeightpoint

\section{1}{Non Bounded Operators on Hilbert Spaces}
A few preliminaries of functional analysis will be needed here.
Let $\cH_1$, $\cH_2$ be complex Hilbert spaces. We consider a linear
operator $T$ defined on a subspace $\Dom T\subset\cH_1$ (called the domain of
$T$) into $\cH_2$. The operator $T$ is said to be {\it densely defined} if
$\Dom T$ is dense in $\cH_1$, and {\it closed} if its graph
$$\Gr\,T=\big\{(x,Tx)~;~x\in\Dom T\big\}$$
is closed in $\cH_1\times\cH_2$.

Assume now that $T$ is closed and densely defined. The adjoint $T^\star$ of $T$ 
(in Von Neumann's sense) is constructed as follows: $\Dom T^\star$ is the set
of $y\in\cH_2$ such that the linear form
$$\Dom T\ni x\longmapsto\langle Tx,y\rangle_2$$
is bounded in $\cH_1$-norm. Since $\Dom T$ is dense, there exists 
for every $y$ in $\Dom T^\star$ a unique element $T^\star y\in\cH_1$ such
that $\langle Tx,y\rangle_2=\langle x,T^\star y\rangle_1$ for all $x\in
\Dom T^\star$. It is immediate to verify that $\Gr\,T^\star=
\big(\Gr(-T)\big)^\perp$ in $\cH_1\times\cH_2$. It follows that
$T^\star$ is closed and that every pair $(u,v)\in\cH_1\times\cH_2$ can be
written
$$(u,v)=(x,-Tx)+(T^\star y,y),~~~~x\in\Dom T,~~y\in\Dom T^\star.$$
Take in particular $u=0$. Then
$$x+T^\star y=0,~~~~v=y-Tx=y+TT^\star y,~~~~
\langle v,y\rangle_2=\|y\|^2_2+\|T^\star y\|_1^2.$$
If $v\in(\Dom T^\star)^\perp$ we get $\langle v,y\rangle_2=0$, thus $y=0$
and $v=0$. Therefore $T^\star$ is densely defined and our discussion implies:

\claim{(1.1) Theorem 
{\rm\cite{Von Neumann 1929})}} If $T:\cH_1\longrightarrow\cH_2$ is
a closed and densely defined operator, then its adjoint $T^\star$ is
also closed and densely defined and $(T^\star)^\star=T$. Furthermore,
we have the relation $\Ker T^\star=(\Im T)^\perp$ and its dual $(\Ker
T)^\perp=\ol{\Im T^\star}$.\qed
\endclaim

Consider now two closed and densely defined operators $T$, $S\,:$
$$\cH_1\buildo T\over\longrightarrow\cH_2\buildo S\over\longrightarrow\cH_3$$
such that $S\circ T=0$. By this, we mean that the range 
$T(\Dom T)$ is contained in $\Ker S\subset \Dom S$, in such a way that
there is no problem for defining the composition $S\circ T$.
The starting point of all $L^2$ estimates is the
following abstract existence theorem.

\claim{(1.2) Theorem} There are orthogonal decompositions
$$\eqalign{
\cH_2&=(\Ker S\cap\Ker T^\star)\oplus\ol{\Im T}
\oplus\ol{\Im S^\star},\cr
\Ker S&=(\Ker S\cap\Ker T^\star)\oplus\ol{\Im T}.\cr}$$
In order that $\Im T=\Ker S$, it suffices that 
$$\|T^\star x\|^2_1+\|Sx\|^2_3\ge C\|x\|^2_2,~~~~
\forall x\in\Dom S\cap\Dom T^\star\leqno(1.3)$$
for some constant $C>0$. In that case, for every $v\in\cH_2$ such that 
$Sv=0$, there exists $u\in\cH_1$ such that $Tu=v$ and 
$$\|u\|_1^2\le{1\over C}\|v\|_2^2.$$
In particular
$$\ol{\Im T}=\Im T=\Ker S,~~~~
\ol{\Im S^\star}=\Im S^\star=\Ker T^\star.$$
\endclaim

\proof{Proof} Since $S$ is closed, the kernel $\Ker S$ is closed in $\cH_2$. The
relation $(\Ker S)^\perp=\ol{\Im S^\star}$ implies
$$\cH_2=\Ker S\oplus\ol{\Im S^\star}\leqno(1.4)$$
and similarly $\cH_2=\Ker T^\star\oplus\ol{\Im T}$. However,
the assumption $S\circ T=0$ shows that $\ol{\Im T}\subset\Ker S$, 
therefore
$$\Ker S=(\Ker S\cap\Ker T^\star)\oplus\ol{\Im T}.\leqno(1.5)$$
The first two equalities in Th.~1.2 are then equivalent to the
conjunction of (1.4) and (1.5).

Now, under assumption (1.3), we are going to show that the equation
$Tu=v$ is always solvable if $Sv=0$. Let $x\in\Dom T^\star$. One can write
$$x=x'+x''~~~~\hbox{\rm where}~~x'\in\Ker S~~\hbox{\rm and}~~x''\in(\Ker S)^\perp
\subset(\Im T)^\perp=\Ker T^\star.$$
Since $x,x''\in\Dom T^\star$, we have also $x'\in\Dom T^\star$. We get
$$\langle v,x\rangle_2=\langle v,x'\rangle_2+\langle v,x''\rangle_2=
\langle v,x'\rangle_2$$
because $v\in\Ker S$ and $x''\in(\Ker S)^\perp$. As $Sx'=0$ and 
$T^\star x''=0$, the Cauchy-Schwarz inequality combined with (1.3) implies
$$|\langle v,x\rangle_2|^2\le\|v\|^2_2~\|x'\|^2_2\le
{1\over C}\|v\|^2_2~\|T^\star x'\|^2_1={1\over C}\|v\|^2_2~\|T^\star x\|^2_1.$$
This shows that the linear form $T^\star_X\ni x\longmapsto\langle x,v\rangle_2$
is continuous on \hbox{$\Im T^\star\subset\cH_1$} with norm
$\le C^{-1/2}\|v\|_2$. By the Hahn-Banach theorem, this form can be extended
to a continuous linear form on $\cH_1$ of norm $\le C^{-1/2}\|v\|_2$, i.e.\
we can find $u\in\cH_1$ such that $\|u\|_1\le C^{-1/2}\|v\|_2$ and
$$\langle x,v\rangle_2=\langle T^\star x,u\rangle_1,~~~~
\forall x\in\Dom T^\star.$$
This means that $u\in\Dom(T^\star)^\star=\Dom T$ and $v=Tu$. We have thus
shown that $\Im T=\Ker S$, in particular $\Im T$ is closed. The dual
\hbox{equality} \hbox{$\Im S^\star=\Ker T^\star$} follows by considering the
dual pair $(S^\star,T^\star)$.\qed
\endproof

\section{2}{Complete Riemannian Manifolds}
Let $(M,g)$ be a riemannian manifold of dimension $m$, with metric
$$g(x)=\sum g_{jk}(x)\,dx_j\otimes dx_k,~~~~1\le j,k\le m.$$
The length of a path $\gamma~:~[a,b]\longrightarrow M$ is by definition
$$\ell(\gamma)=\int_a^b|\gamma'(t)|_gdt=\int_a^b\Big(
\sum_{j,k}g_{jk}\big(\gamma(t)\big)\,\gamma'_j(t)\gamma'_k(t)\Big)^{1/2}dt.$$
The geodesic distance of two points $x,y\in M$ is
$$\delta(x,y)=\inf_\gamma\ell(\gamma)~~~~\hbox{\rm over paths $\gamma$ with}
~~\gamma(a)=x,~~\gamma(b)=y,$$
if $x,y$ are in the same connected component of $M$, $\delta(x,y)=+\infty$
otherwise. It~is easy to check that $\delta$ satisfies the usual axioms
of distances: for the separation axiom, use the fact that if $y$ is outside
some closed coordinate ball $\ol B$ of radius $r$ centered at $x$ and if
$g\ge c|dx|^2$ on $\ol B$, then $\delta(x,y)\ge c^{1/2}r$. In~addition,
$\delta$ satisfies the axiom:
$$\hbox{for every $x,y\in M$,}~~~~\inf_{z\in M}
\max\{\delta(x,z),\delta(y,z)\}={1\over 2}\delta(x,y).\qquad\leqno(2.1)$$
In fact for every $\varepsilon>0$ there is a path $\gamma$ such that
$\gamma(a)=x$, $\gamma(b)=y$, \hbox{$\ell(\gamma)<\delta(x,y)+\varepsilon$}
and we can take $z$ to be at mid-distance between $x$ and $y$ along~$\gamma$.
A metric space $E$ with a distance $\delta$ satisfying the additional
axiom~(2.1) will be called a {\it geodesic} metric space. It is then easy
to see by dichotomy that any two points $x,y\in E$ can be joined by a chain
of points \hbox{$x=x_0$}, \hbox{$x_1,\ldots,x_N=y$} such that
$\delta(x_j,x_{j+1})<\varepsilon$ and
\hbox{$\sum\delta(x_j,x_{j+1})<\delta(x,y)+\varepsilon$}.

\claim{(2.2) Lemma {\rm(Hopf-Rinow)}} Let $(E,\delta)$ be a geodesic
metric space. Then the following properties are equivalent:
\smallskip
\item{\rm a)} $E$ is locally compact and complete$\,;$
\smallskip
\item{\rm b)} all closed geodesic balls $\ol B(x_0,r)$ are compact.
\endclaim

\proof{Proof} Since any Cauchy sequence is bounded, it is immediate that b)
implies~a). We now check that a)~$\Longrightarrow$~b). Fix $x_0$ and
define $R$ to be the supremum of all $r>0$ such that $\ol B(x_0,r)$ is
compact. Since $E$ is locally compact, we have $R>0$. Suppose that
$R<+\infty$. Then $\ol B(x_0,r)$ is compact for every~$r<R$. Let
$y_\nu$ be a sequence of points in $\ol B(x_0,R)$. Fix an integer~$p$.
As $\delta(x_0,y_\nu)\le R$, axiom~(2.1) shows that we can find points
$z_\nu\in M$ such that $\delta(x_0,z_\nu)\le(1-2^{-p})R$ and
$\delta(z_\nu,y_\nu)\le 2^{1-p}R$. Since
$\ol B(x_0,(1-2^{-p})R)$ is compact, there is a subsequence
$(z_{\nu(p,q)})_{q\in\bN}$ converging to a limit
point~$w_p$ with $\delta(z_{\nu(p,q)},w_p)\le 2^{-q}$.
We proceed by induction on $p$ and take $\nu(p+1,q)$ to be a
subsequence of~$\nu(p,q)$. Then
$$\delta(y_{\nu(p,q)},w_p)\le
\delta(y_{\nu(p,q)},z_{\nu(p,q)})+\delta(z_{\nu(p,q)},w_p)
\le 2^{1-p}R+2^{-q}.$$
Since $(y_{\nu(p+1,q)})$ is a subsequence of $(y_{\nu(p,q)})$, we infer
that \hbox{$\delta(w_p,w_{p+1})\le 3\,2^{-p}R$} by letting $q$
tend to~$+\infty$. By the completeness hypo\-thesis, the Cauchy sequence
$(w_p)$ converges to a limit point $w\in M$, and the above inequalities
show that $(y_{\nu(p,p)})$ converges to~$w\in\ol B(x_0,R)$. Therefore
$\ol B(x_0,R)$ is compact. Now, each point $y\in\ol B(x_0,R)$
can be covered by a compact ball $\ol B(y,\varepsilon_y)$, and the
compact set $\ol B(x_0,R)$ admits a finite covering
by concentric balls $B(y_j,\varepsilon_{y_j}/2)$. Set $\varepsilon=\min
\varepsilon_{y_j}$. Every point $z\in\ol B(x_0,R+\varepsilon/2)$ is at
distance $\le\varepsilon/2$ of some point $y\in\ol B(x_0,R)$, hence
at distance $\le\varepsilon/2+\varepsilon_{y_j}/2$ of some point~$y_j$,
in particular $\ol B(x_0,R+\varepsilon/2)\subset
\bigcup\ol B(y_j,\varepsilon_{y_j})$ is compact. This is a contradiction,
so~$R=+\infty$.\qed
\endproof
        
The following standard definitions and properties will be useful in order
to deal with the completeness of the metric.

\claim{(2.3) Definitions} \smallskip
\item{\rm a)} A riemannian manifold $(M,g)$ is said to be complete 
if $(M,\delta)$ is complete as a metric space.
\smallskip
\item{\rm b)} A continuous function $\psi~:~M\to\bR$ is said to be exhaustive
if for every $c\in\bR$ the sublevel set $M_c=\{x\in M~;~\psi(x)<c\}$
is relatively compact in~$M$.
\smallskip
\item{\rm c)} A sequence $(K_\nu)_{\nu\in\bN}$ of compact subsets of
$M$ is said to be exhaustive if $M=\bigcup K_\nu$ and if $K_\nu$ is contained 
in the interior of $K_{\nu+1}$ for all $\nu$ $($so that every compact subset
of $M$ is contained in some $K_\nu)$.\smallskip
\endclaim

\claim{(2.4) Lemma} The following properties are equivalent:
\smallskip
\item{\rm a)} $(M,g)$ is complete;
\smallskip
\item{\rm b)} there exists an exhaustive function $\psi\in \cC^\infty(M,\bR)$ such
that $|d\psi|_g\le1\,;$
\smallskip
\item{\rm c)} there exists an exhaustive sequence $(K_\nu)_{\nu\in\bN}$
of compact subsets of $M$ and functions $\psi_\nu\in \cC^\infty(M,\bR)$ such that
$$\eqalign{
&\psi_\nu=1~~~\hbox{\rm in~a~neighborhood~of}~K_\nu,~~~~
\Supp\,\psi_\nu\subset K^\circ_{\nu+1},\cr
&0\le\psi_\nu\le 1~~~\hbox{\rm and}~~|d\psi_\nu|_g\le 2^{-\nu}.\cr}$$
\endclaim

\proof{Proof} a) $\Longrightarrow$ b). Without loss of generality, we may 
assume that $M$ is connected. Select a point $x_0\in M$ and set $\psi_0(x)=
{1\over 2}\delta(x_0,x)$. Then $\psi_0$ is a Lipschitz function with constant 
${1\over 2}$, thus $\psi_0$ is differentiable almost everywhere on $M$
and $|d\psi_0|_g\le{1\over 2}$. We can find a smoothing $\psi$ of $\psi_0$
such that $|d\psi|_g\le 1$ and $|\psi-\psi_0|\le 1$. Then $\psi$ is an
exhaustion function of $M$.
\medskip
\noindent b) $\Longrightarrow$ c). Choose $\psi$ as in a) and a function 
$\rho\in \cC^\infty(\bR,\bR)$ such that $\rho=1$ on $]-\infty,1.1]$,
$\rho=0$ on $[1.9,+\infty[$ and $0\le\rho'\le 2$ on $[1,2]$. Then
$$K_\nu=\{x\in M~;~\psi(x)\le 2^{\nu+1}\},~~~~\psi_\nu(x)=
\rho\big(2^{-\nu-1}\psi(x)\big)$$
satisfy our requirements.
\medskip
\noindent c) $\Longrightarrow$ b). Set $\psi=\sum 2^\nu(1-\psi_\nu)$.
\medskip
\noindent b) $\Longrightarrow$ a). The inequality $|d\psi|_g\le 1$ implies
$|\psi(x)-\psi(y)|\le\delta(x,y)$ for all $x,y\in M$, so all
$\delta$-balls must be relatively compact in $M$.\qed
\endproof

\section{3}{$L^2$ Hodge Theory on Complete Riemannian Manifolds}
Let $(M,g)$ be a riemannian manifold and let $F_1,F_2$ be hermitian $\cC^\infty$
vector bundles over $M$. If $P:\cC^\infty(M,F_1)\longrightarrow \cC^\infty(M,F_2)$ is a differential 
operator with smooth coefficients, then $P$ induces a non bounded operator
$$\wtP:L^2(M,F_1)\longrightarrow L^2(M,F_2)$$
as follows: if $u\in L^2(M,F_1)$, we compute $\wtP u$
in the sense of distribution theory and we say that $u\in\Dom\wtP$ if 
$\wtP u\in L^2(M,F_2)$. It follows that $\wtP$ is densely defined, since 
$\Dom P$ contains the set $\cD(M,F_1)$ of compactly supported sections of
$\cC^\infty(M,F_1)$, which is dense in $L^2(M,F_1)$. Furthermore $\Gr\,\wtP$ is
closed: if $u_\nu\to u$ in $L^2(M,F_1)$ and $\wtP u_\nu\to v$ in $L^2(M,F_2)$
then $\wtP u_\nu\to\wtP u$ in the weak topology of distributions, thus we must
have $\wtP u=v$ and $(u,v)\in\Gr\,\wtP$. By the general results of
\S~1, we see that $\wtP$ has a closed and densely defined Von Neumann 
adjoint $\big(\wtP\big)^\star$. We want to stress, however, that
$\big(\wtP\big)^\star$ {\it does not always} coincide
with the extension $(P^\star)^{\sim}$ of the formal adjoint
$P^\star:\cC^\infty(M,F_2)\longrightarrow \cC^\infty(M,F_1)$, computed in the sense of
distribution theory.  In fact $u\in\Dom(\wtP)^\star$, resp.\
$u\in\Dom(P^\star)^{\sim}$, if and only if there is an element
$v\in L^2(M,F_1)$ such that $\langle u,\wtP f\rangle=
\langle v,f\rangle$ for all $f\in\Dom\wtP$, resp.\ for all
$f\in\cD(M,F_1)$. Therefore we always have
$\Dom(\wtP)^\star\subset\Dom(P^\star)^{\sim}$ and the inclusion
may be strict because the integration by parts
to perform may involve boundary integrals for~$(\wtP)^\star$.

\claim{(3.1) Example} \rm Consider
$$P={d\over dx}:L^2\big(]0,1[\big)\longrightarrow L^2\big(]0,1[\big)$$
where the $L^2$ space is taken with respect to the Lebesgue measure~$dx$.
Then $\Dom\wtP$ consists of all $L^2$ functions with $L^2$ derivatives
on~$]0,1[$. Such functions have a continuous extension to the
interval~$[0,1]$. An integration by parts shows that
$$\int_0^1 u\ol{df\over dx}\,dx=\int_0^1 -{du\over dx}\ol f\,dx$$
for all $f\in\cD(]0,1[)$, thus $P^\star=-d/dx=-P$. However for
$f\in\Dom\wtP$ the integration by parts involves the extra term
$u(1)\ol f(1)-u(0)\ol f(0)$ in the right hand side,
which is thus continuous in $f$ with respect to the $L^2$ topology
if and only if $du/dx\in L^2$ and $u(0)=u(1)=0$.
Therefore $\Dom(\wtP)^\star$ consists of all $u\in\Dom(P^\star)^\sim=\Dom\wtP$
satisfying the additional boundary condition $u(0)=u(1)=0$.\qed
\endclaim

Let $E\to M$ be a differentiable hermitian bundle. In what follows, we 
still denote by $D,\delta,\Delta$ the differential operators of 
\S~VI-2 extended in the sense of distribution theory (as explained above).
These operators are thus closed and densely defined operators
on $L_\bu^2(M,E)=\bigoplus_pL_p^2(M,E)$. We also introduce the space
$\cD_p(M,E)$ of compactly supported forms in $\cC^\infty_p(M,E)$. The theory
relies heavily on the following important result.

\claim{(3.2) Theorem} Assume that $(M,g)$ is complete. Then
\smallskip
\item{\rm a)} $\cD_\bu(M,E)$ is dense in $\Dom D$, $\Dom\delta$ and
$\Dom D\cap\Dom\delta$ respectively for the graph norms
$$u\mapsto\|u\|+\|Du\|,~~~~u\mapsto\|u\|+\|\delta u\|,~~~~
  u\mapsto\|u\|+\|Du\|+\|\delta u\|.$$
\smallskip
\item{\rm b)} $D^\star=\delta$, $\delta^\star=D$ as adjoint operators
in Von Neumann's sense.
\smallskip
\item{\rm c)} One has $\langle u,\Delta u\rangle=\|Du\|^2+
\|\delta u\|^2$ for every $u\in\Dom\Delta$. In particular
$$\Dom\Delta\subset\Dom D\cap\Dom\delta,~~~~\Ker\Delta=\Ker D\cap\Ker\delta,$$
and $\Delta$ is self-adjoint.
\smallskip
\item{\rm d)} If $D^2=0$, there are orthogonal decompositions
$$\eqalign{
L^2_\bu(M,E)&=\cH^\bu(M,E)\oplus\ol{\Im D}\oplus\ol{\Im \delta},\cr
\Ker D&=\cH^\bu(M,E)\oplus\ol{\Im D},\cr}$$
\vskip-3pt\item{} where
$\cH^\bu(M,E)=\big\{u\in L^2_\bu(M,E)\,;~\Delta u=0\big\}\subset
\cC^\infty_\bu(M,E)$ is the space of $L^2$ harmonic forms.\smallskip
\endclaim

\proof{Proof} a) We show that every element $u\in\Dom D$ can be approximated
in the graph norm of $D$ by smooth and compactly supported forms.  By
hypothesis, $u$ and $Du$ belong to $L^2_\bu(M,E)$.  Let $(\psi_\nu)$ be
a sequence of functions as in Lemma~2.4~c).  Then $\psi_\nu u\to u$
in $L^2_\bu(M,E)$ and $D(\psi_\nu u)=\psi_\nu Du+d\psi_\nu\wedge u$
where
$$|d\psi_\nu\wedge u|\le|d\psi_\nu|~|u|\le 2^{-\nu}|u|.$$
Therefore $d\psi_\nu\wedge u\to 0$ and $D(\psi_\nu u)\to Du$.  After
replacing $u$ by $\psi_\nu u$, we may assume that $u$ has compact
support, and by using a finite partition of unity on a neighborhood of 
$\Supp\,u$ we may also assume that $\Supp\,u$ is contained 
in a coordinate chart of $M$ on which $E$ is trivial.  Let $A$ be the
connection form of $D$ on this chart and
$(\rho_\varepsilon)$ a family of smoothing kernels. Then
$u\star\rho_\varepsilon\in\cD_\bu(M,E)$ converges to $u$ in $L^2(M,E)$ and
$$D(u\star\rho_\varepsilon)-(Du)\star\rho_\varepsilon=
A\wedge(u\star\rho_\varepsilon)-(A\wedge u)\star\rho_\varepsilon$$
because $d$ commutes with convolution (as any differential operator with
constant coefficients). Moreover $(Du)\star\rho_\varepsilon$
converges to $Du$ in $L^2(M,E)$ and $A\wedge(u\star\rho_\varepsilon)$,
$(A\wedge u)\star\rho_\varepsilon$ both converge to $A\wedge u$
since $A\wedge{\scriptstyle\bu}$
acts continuously on $L^2$. Thus $D(u\star\rho_\varepsilon)$
converges to $Du$ and the density of $\cD_\bu(M,E)$ in $\Dom D$ follows. 
The proof for $\Dom\delta$ and $\Dom D\cap\Dom\delta$ is similar, except
that the principal part of $\delta$
no longer has constant coefficients in general. The convolution
technique requires in this case the following lemma due to
K.O.~Friedrichs (see e.g.\ \cite{H\"ormander~1963}).
\endproof

\claim{(3.3) Lemma} Let $Pf=\sum a_k\,\partial f/\partial x_k+bf$ be a
differential operator of order~$1$ on an open set $\Omega\subset\bR^n$,
with coefficients $a_k\in C^1(\Omega)$, $b\in C^0(\Omega)$. Then for any
$v\in L^2(\bR^n)$ with compact support in~$\Omega$ we have
$$\lim_{\varepsilon\to0}
||P(v\star\rho_\varepsilon)-(Pv)\star\rho_\varepsilon||_{L^2}=0.$$
\endclaim

\proof{Proof} It is enough to consider the case when $P=a\partial/\partial x_k$.
As the result is obvious if $v\in C^1$, we only have to show that
$$||P(v\star\rho_\varepsilon)-(Pv)\star\rho_\varepsilon||_{L^2}\le
C||v||_{L^2}$$
and to use a density argument. A computation of
$w_\varepsilon=P(v\star\rho_\varepsilon)-(Pv)\star\rho_\varepsilon$
by means of an integration by parts gives
$$\eqalign{
w_\varepsilon(x)
&=\int_{\bR^n}\Big(a(x){\partial v\over\partial x_k}
(x-\varepsilon y)\rho(y)-a(x-\varepsilon y){\partial v\over\partial x_k}
(x-\varepsilon y)\rho(y)\Big)dy\cr
&=\int_{\bR^n}\Big(\big(a(x)-a(x-\varepsilon y)\big)
v(x-\varepsilon y){1\over\varepsilon}\partial_k\rho(y)\cr
&\qquad\qquad\qquad\qquad\qquad{}+
\partial_ka(x-\varepsilon y)v(x-\varepsilon y)\rho(y)\Big)dy.\cr}$$
If $C$ is a bound for $|da|$ in a neighborhood of $\Supp\,v$, we get
$$|w_\varepsilon(x)|\le C\int_{\bR^n}|v(x-\varepsilon y)|\big(
|y|\,|\partial_k\rho(y)|+|\rho(y)|\big)dy,$$
so Minkowski's inequality $||f\star g||_{L^p}\le||f||_{L^1}||g||_{L^p}$
gives
$$||w_\varepsilon||_{L^2}\le C\Big(\int_{\bR^n}\big(|y|\,|\partial_k\rho(y)|
+|\rho(y)|\big)dy\Big)||v||_{L^2}.\eqno\square$$
\endproof

\proof{Proof (end).} b) is equivalent to the fact that
$$\Ll Du,v\Gg=\Ll u,\delta v\Gg,~~~~\forall u\in\Dom D,~~
\forall v\in\Dom\delta.$$
By a), we can find $u_\nu,~v_\nu\in\cD_\bu(M,E)$ such that 
$$u_\nu\to u,~~~~v_\nu\to v,~~~~Du_\nu\to Du~~~~\hbox{\rm and}~~\delta v_\nu\to
\delta v~~~\hbox{\rm in}~~L^2_\bu(M,E),$$
and the required equality is the limit of the equalities
$\Ll Du_\nu,v_\nu\Gg=\Ll u_\nu,\delta v_\nu\Gg$.
\medskip
\noindent c) Let $u\in\Dom\Delta$. As $\Delta$ is an elliptic operator of
order $2$, $u$ must be in $W^2_\bu(M,E,\loc)$ by
G\aa rding's inequality. In particular $Du,~\delta u\in L^2(M,E,\loc)$
and we can perform all integrations by parts that we want if the forms
are multiplied by compactly supported functions $\psi_\nu$. Let us compute
$$\leqalignno{
&\|\psi_\nu Du\|^2+\|\psi_\nu\delta u\|^2=\cr
&=\Ll\psi_\nu^2Du,Du\Gg+\Ll u,D(\psi_\nu^2\delta u)\Gg\cr
&=\Ll D(\psi_\nu^2u),Du\Gg+\Ll u,\psi_\nu^2D\delta u\Gg-
2\Ll\psi_\nu d\psi_\nu\wedge u,Du\Gg+
2\Ll u,\psi_\nu d\psi_\nu\wedge\delta u\Gg\cr
&=\Ll\psi_\nu^2u,\Delta u\Gg-2\Ll d\psi_\nu\wedge u,\psi_\nu Du\Gg
+2\Ll u,d\psi_\nu\wedge(\psi_\nu\delta u)\Gg\cr
&\le\Ll\psi_\nu^2u,\Delta u\Gg+2^{-\nu}\big(2\|\psi_\nu Du\|~\|u\|+
2\|\psi_\nu\delta u\|~\|u\|\big)\cr
&\le\Ll\psi_\nu^2u,\Delta u\Gg+2^{-\nu}\big(\|\psi_\nu Du\|^2+
\|\psi_\nu\delta u\|^2+2\|u\|^2\big).\cr}$$
We get therefore
$$\|\psi_\nu Du\|^2+\|\psi_\nu\delta u\|^2\le{1\over 1-2^{-\nu}}
\big(\Ll\psi_\nu^2u,\Delta u\Gg+2^{1-\nu}\|u\|^2\big).$$
By letting $\nu$ tend to $+\infty$, we obtain $\|Du\|^2+\|\delta u\|^2\le
\Ll u,\Delta u\Gg$, in particular $Du$, $\delta u$ are in $L^2_\bu(M,E)$.
This implies
$$\Ll u,\Delta v\Gg=\Ll Du,Dv\Gg+\Ll\delta u,\delta v\Gg,~~~~\forall u,v
\in\Dom\Delta,$$
because the equality holds for $\psi_\nu u$ and $v$, and because we have
\hbox{$\psi_\nu u\to u$}, \hbox{$D(\psi_\nu u)\to Du$} and
\hbox{$\delta(\psi_\nu u)\to\delta u$} in $L^2$.
Therefore $\Delta$ is self-adjoint.
\medskip
\noindent d) is an immediate consequence of b), c) and Th.~1.2.\qed
\endproof

On a complete hermitian manifold $(X,\omega)$, there are of course
similar results for the operators $D',D'',\delta',\delta'',\Delta',\Delta''$
attached to a hermitian vector bundle $E$.

\section{4}{General Estimate for $d''$ on Hermitian Manifolds} 
Let $(X,\omega)$ be a {\it complete} hermitian manifold and $E$ a hermitian 
holomorphic vector bundle of rank $r$ over $X$. Assume that the hermitian
operator
$$A_{E,\omega}=[\ii\Theta(E),\Lambda]+T_\omega\leqno(4.1)$$
is {\it semi-positive} on $\Lambda^{p,q}T^\star_X\otimes E$. Then for every 
form $u\in\Dom D''\cap\Dom \delta''$ of bidegree $(p,q)$ we have
$$\|D''u\|^2+\|\delta''u\|^2\ge\int_X\langle A_{E,\omega}u,u\rangle\,dV.
\leqno(4.2)$$
In fact (4.2) is true for all $u\in\cD_{p,q}(X,E)$ in view of the
Bochner-Kodaira-Nakano identity VII-2.3, and this result is easily extended
to every $u$ in $\Dom D''\cap\Dom \delta''$ by density of $\cD_{p,q}(X,E)$
(Th.~3.2~a)).

Assume now that a form $g\in L^2_{p,q}(X,E)$ is given such that
$$D''g=0,\leqno(4.3)$$
and that for almost every $x\in X$ there exists $\alpha\in[0,+\infty[$ such that
$$|\langle g(x),u\rangle|^2\le\alpha\,\langle A_{E,\omega}u,u\rangle$$
for every $u\in(\Lambda^{p,q}T^\star_X\otimes E)_x$. If the operator
$A_{E,\omega}$ is invertible, the minimal such number $\alpha$ is
$|A_{E,\omega}^{-1/2}g(x)|^2=\langle A_{E,\omega}^{-1}g(x),g(x)\rangle$,
so we shall always denote it in this way even when $A_{E,\omega}$ is
no longer invertible. Assume furthermore that
$$\int_X\langle A_{E,\omega}^{-1}g,g\rangle\,dV<+\infty.\leqno(4.4)$$
The basic result of $L^2$ theory can be stated as follows.

\claim{(4.5) Theorem} If $(X,\omega)$ is complete and $A_{E,\omega}\ge 0$
in bidegree $(p,q)$, then for any $g\in L^2_{p,q}(X,E)$ satisfying
{\rm (4.4)} such that $D''g=0$ there exists $f\in L^2_{p,q-1}(X,E)$ such that 
$D''f=g$ and
$$\|f\|^2\le\int_X\langle A_{E,\omega}^{-1}g,g\rangle\,dV.$$
\endclaim

\proof{Proof} For every $u\in\Dom D''\cap\Dom \delta''$ we have
$$\eqalign{
\big|\Ll u,g\Gg\big|^2=\Big|\int_X\langle u,g\rangle\,dV\Big|^2
&\le\Big(\int_X\langle A_{E,\omega}u,u\rangle^{1/2}
\langle A_{E,\omega}^{-1}g,g\rangle^{1/2}\,dV\Big)^2\cr
&\le\int_X\langle A_{E,\omega}^{-1}g,g\rangle\,dV\cdot
    \int_X\langle A_{E,\omega}u,u\rangle\,dV\cr}$$
by means of the Cauchy-Schwarz inequality. The a priori estimate (4.2) implies
$$\big|\Ll u,g\Gg\big|^2\le C\big(\|D''u\|^2+\|\delta''u\|^2\big),~~~~
\forall u\in\Dom D''\cap\Dom\delta''$$
where $C$ is the integral (4.4). Now we just have to repeat the proof of the
existence part of Th.~1.2. For any $u\in\Dom\delta''$, let us write
$$u=u_1+u_2,~~~~u_1\in\Ker D'',~~~~u_2\in(\Ker D'')^\perp=
\ol{\Im \delta''}.$$
Then $D''u_1=0$ and $\delta''u_2=0$. Since $g\in\Ker D''$, we get
$$\big|\Ll u,g\Gg\big|^2=\big|\Ll u_1,g\Gg\big|^2\le C\|\delta''u_1\|^2
=C\|\delta''u\|^2.$$
The Hahn-Banach theorem shows that the continuous linear form
$$L^2_{p,q-1}(X,E)\ni\delta''u\longmapsto\Ll u,g\Gg$$
can be extended to a linear form $v\longmapsto\Ll v,f\Gg$, 
$f\in L^2_{p,q-1}(X,E)$, of norm $\|f\|\le C^{1/2}$. This means that
$$\Ll u,g\Gg=\Ll\delta''u,f\Gg,~~~~\forall u\in\Dom\delta'',$$
i.e.\ that $D''f=g$. The theorem is proved.\qed
\endproof

\claim{(4.6) Remark} \rm One can always find a solution $f\in(\Ker D'')^\perp\,:$
otherwise replace f by its orthogonal projection on $(\Ker D'')^\perp$. 
This solution is clearly unique and is precisely the solution of minimal
$L^2$ norm of the equation $D''f=g$.  We have $f\in\ol{\Im\delta''}$,
thus $f$ satifies the additional equation
$$\delta''f=0.\leqno(4.7)$$
Consequently $\Delta''f=\delta''D''f=\delta''g$. If $g\in \cC^\infty_{p,q}(X,E)$,
the ellipticity of $\Delta''$ shows that $f\in \cC^\infty_{p,q-1}(X,E)$.
\endclaim

\claim{(4.8) Remark} \rm If $A_{E,\omega}$ is positive definite, let
$\lambda(x)>0$ be the smallest eigenvalue of this operator at $x\in X$.
Then $\lambda$ is continuous on $X$ and we have
$$\int_X\langle A_{E,\omega}^{-1}g,g\rangle\,dV\le\int_X\lambda(x)^{-1}
|g(x)|^2\,dV.$$
The above situation occurs for example if $\omega$ is complete K\"ahler,
$E>_m0$ and $p=n$, $q\ge 1$, $m\ge\min\{n-q+1,r\}$ (apply
Lemma~VII-7.2).
\endclaim

\section{5}{Estimates on Weakly Pseudoconvex Manifolds}
We first introduce a large class of complex manifolds on which the $L^2$
estimates will be easily tractable.

\claim{(5.1) Definition} A complex manifold $X$ is said to be weakly 
pseudo\-convex if there exists an exhaustion function $\psi\in \cC^\infty(X,\bR)$
such that $\ii d'd''\psi\ge 0$ on $X$, i.e.\ $\psi$ is plurisubharmonic.
\endclaim

For domains $\Omega\subset\bC^n$, the above weak pseudoconvexity notion
is equivalent to pseudoconvexity (cf.\ Th.~I-4.14).
Note that every compact manifold is also weakly pseudoconvex (take 
$\psi\equiv 0$). Other examples that will appear later are Stein manifolds,
or the total space of a Griffiths semi-negative vector bundle over a compact 
manifold (cf.\ Prop.~IX-?.?).

\claim{(5.2) Theorem} Every weakly pseudoconvex K\"ahler manifold $(X,\omega)$
carries a complete K\"ahler metric $\wh\omega$.
\endclaim

\proof{Proof} Let $\psi\in \cC^\infty(X,\bR)$ be an exhaustive plurisubharmonic
function on $X$. After addition of a constant to $\psi$, we can assume 
$\psi\ge 0$. Then $\wh\omega=\omega+\ii d'd''(\psi^2)$ is K\"ahler and
$$\wh\omega=\omega+2i\psi d'd''\psi+2\ii d'\psi\wedge d''\psi\ge
\omega+2\ii d'\psi\wedge d''\psi.$$
Since $d\psi=d'\psi+d''\psi$, we get $|d\psi|_{\wh\omega}=\sqrt{2}
|d'\psi|_{\wh\omega}\le 1$ and Lemma~2.4 shows that $\wh\omega$ is 
complete.\qed
\endproof

Observe that we could have set more generally $\wh\omega=\omega+\ii d'd''
(\chi\circ\psi)$ where $\chi$ is a convex increasing function. Then
$$\leqalignno{
\wh\omega&=\omega+i(\chi'\circ\psi)d'd''\psi+i(\chi''\circ\psi)d'\psi\wedge
d''\psi\cr
&\ge\omega+\ii d'(\rho\circ\psi)\wedge d''(\rho\circ\psi)&(5.3)\cr}$$
where $\rho(t)=\int_0^t\sqrt{\chi''(u)}\,du$. We thus have
$|d'(\rho\circ\psi)|_{\wh\omega}\le 1$ and $\wh\omega$ will be
complete as soon as $\lim_{t\to+\infty}\rho(t)=+\infty$, i.e.\
$$\int_0^{+\infty}\sqrt{\chi''(u)}\,du=+\infty.\leqno(5.4)$$
One can take for example $\chi(t)=t-\log(t)$ for $t\ge 1$.

It follows from the above considerations that almost all vanishing theorems
for positive vector bundles over compact manifolds are also valid on
weakly pseudoconvex manifolds. Let us mention here the analogues of some
results proved in Chapter~7.

\claim{(5.5) Theorem} For any $m$-positive vector bundle of rank $r$ over a 
weakly pseudoconvex manifold $X$, we have $H^{n,q}(X,E)=0$ for all
$q\ge 1$ and \hbox{$m\ge\min\{n-q+1,r\}$}.
\endclaim

\proof{Proof} The curvature form $\ii\Theta(\det E)$ is a K\"ahler metric
on $X$, hence $X$ possesses a complete K\"ahler metric $\omega$. Let 
$\psi\in \cC^\infty(X,\bR)$ be an exhaustive plurisubharmonic
function. For any convex increasing function $\chi\in \cC^\infty(\bR,\bR)$,
we denote by $E_\chi$ the holomorphic vector bundle $E$ together with the 
modified metric $|u|_\chi^2=|u|^2\,\exp\big(-\chi\circ\psi(x)\big)$, 
$u\in E_x$. We get
$$\ii\Theta(E_\chi)=\ii\Theta(E)+\ii d'd''(\chi\circ\psi)\otimes\,\Id_E
\ge_m \ii\Theta(E),$$
thus $A_{E_\chi,\omega}\ge A_{E,\omega}>0$ in bidegree $(n,q)$. Let $g$ be a
given form of bidegree $(n,q)$ with $L^2_\loc$ coefficients, such that
$D''g=0$. The integrals
$$\int_X\langle A_{E_\chi,\omega}^{-1}g,g\rangle_\chi\,dV\le
\int_X\langle A_{E,\omega}^{-1}g,g\rangle\,e^{-\chi\circ\psi}\,dV,~~~~
\int_X |g|^2\,e^{-\chi\circ\psi}\,dV$$
become convergent if $\chi$ grows fast enough. We can thus apply Th.~4.5
to $(X,E_\chi,\omega)$ and find a $(n,q-1)$ form $f$ such that $D''f=g$. 
If $g$ is smooth, Remark~4.6 shows that $f$ can also be chosen smooth.\qed
\endproof

\claim{(5.6) Theorem} If $E$ is a positive line bundle over a weakly
pseudoconvex manifold $X$, then $H^{p,q}(X,E)=0$ for
$p+q\ge n+1$.
\endclaim

\proof{Proof} The proof is similar to that of Th.~5.5, except that
we use here the K\"ahler metric 
$$\omega_\chi=\ii\Theta(E_\chi)=\omega+\ii d'd''(\chi\circ\psi),~~~~
\omega=\ii\Theta(E),$$
which depends on $\chi$. By (5.4) $\omega_\chi$ is complete as soon as
$\chi$ is a convex increasing function that grows fast enough. Apply
now Th.~4.5 to $(X,E_\chi,\omega_\chi)$ and observe that
$A_{E_\chi,\omega_\chi}=[\ii\Theta(E_\chi),\Lambda_\chi]=
(p+q-n)\,\Id$~ in bidegree $(p,q)$ in virtue of Cor.~VI-8.4 It remains
to show that for every form $g\in \cC^\infty_{p,q}(X,E)$ there exists a choice
of $\chi$ such that $g\in L^2_{p,q}(X,E_\chi,\omega_\chi)$. By (5.3)
the norm of a scalar form with respect to $\omega_\chi$ is less than
its norm with respect to $\omega$, hence
$|g|_\chi^2\le|g|^2\,\exp(-\chi\circ\psi)$. On the other hand
$$dV_\chi\le C\big(1+\chi'\circ\psi+\chi''\circ\psi\big)^n\,dV$$
where $C$ is a positive continuous function on $X$. The following lemma
implies that we can always choose $\chi$ in order that the integral of 
$|g|_\chi^2\,dV_\chi$ converges on $X$.
\endproof

\claim{(5.7) Lemma} For any positive function $\lambda\in \cC^\infty\big(
[0,+\infty[,\bR\big)$, there exists a smooth convex function
$\chi\in \cC^\infty\big([0,+\infty[,\bR\big)$
such that $\chi,\chi',\chi''\ge\lambda$ and
\hbox{$(1+\chi'+\chi'')^ne^{-\chi}\le 1/\lambda$}.
\endclaim

\proof{Proof} We shall construct $\chi$ such that $\chi''\ge
\chi'\ge\chi\ge\lambda$ and \hbox{$\chi''/\chi^2\le C$}
for some constant $C$. Then $\chi$ satisties the conclusion of the
lemma after addition of a constant. Without loss of generality,
we may assume that $\lambda$ is increasing and $\lambda\ge 1$. 
We define $\chi$ as a power series
$$\chi(t)=\sum_{k=0}^{+\infty}\,a_0a_1\ldots a_k\,t^k,$$
where $a_k>0$ is a decreasing sequence converging to $0$ very slowly.
Then $\chi$ is real analytic on $\bR$ and the inequalities
$\chi''\ge\chi'\ge\chi$ are realized if we choose $a_k\ge 1/k$, $k\ge1$.
Select a strictly increasing sequence of integers $(N_p)_{p\ge 1}$
so large that ${1\over p}\lambda(p+1)^{1/N_p}\in[1/p,1/(p-1)[$.
We set
$$\eqalign{
a_0&=\ldots=a_{N_1-1}=e\,\lambda(2),\cr
a_k&={1\over p}\,\lambda(p+1)^{1/N_p}\,e^{1/\sqrt{k}},~~~~N_p\le k<N_{p+1}.
\cr}$$
Then $(a_k)$ is decreasing. For $t\in[0,1]$ we have $\chi(t)\ge a_0
\ge\lambda(t)$ and for $t\in[1,+\infty[$ the choice $k=N_p$ where
$p=[t]$ is the integer part of $t$ gives
$$\chi(t)\ge\chi(p)\ge(a_0a_1\ldots a_k)p^k\ge(a_kp)^k\ge\lambda(p+1)\ge
\lambda(t).$$
Furthermore, we have
$$\eqalign{
\chi(t)^2&\ge\sum_{k\ge 0}~(a_0a_1\ldots a_k)^2\,t^{2k},\cr
\chi''(t)=&\sum_{k\ge 0}~(k+1)(k+2)\,a_0a_1\ldots a_{k+2}\,t^k,\cr}$$
thus we will get $\chi''(t)\le C\chi(t)^2$ if we can prove that
$$m^2\,a_0a_1\ldots a_{2m}\le C'(a_0a_1\ldots a_m)^2,~~~~m\ge 0.$$
However, as ${1\over p}\lambda(p+1)^{1/N_p}$ is decreasing, we find
$$\eqalignno{
{a_0a_1\ldots a_{2m}\over(a_0a_1\ldots a_m)^2}&=
{a_{m+1}\ldots a_{2m}\over a_0a_1\ldots a_m}\cr
&\le\exp\Big({1\over\sqrt{m+1}}+\cdots+{1\over\sqrt{2m}}-
{1\over\sqrt{1}}-\cdots-{1\over\sqrt{m}}+O(1)\Big)\cr
&\le\exp\big(2\sqrt{2m}-4\sqrt{m}+O(1)\big)\le C'm^{-2}.&\square\cr}$$
\endproof

As a last application, we generalize the Girbau vanishing theorem
in the case of weakly pseudoconvex manifolds. This result is due to
\cite{Abdelkader~1980} and \cite{Ohsawa~1981}. We present here a simplified 
proof which appeared in \cite{Demailly~1985}.

\claim{(5.8) Theorem} Let $(X,\omega)$ be a weakly pseudoconvex K\"ahler
manifold. If $E$ is a semi-positive line bundle such that $\ii\Theta(E)$ has at
least $n-s+1$ positive eigenvalues at every point, then
$$H^{p,q}(X,E)=0~~~~\hbox{\rm for}~~p+q\ge n+s.$$
\endclaim

\proof{Proof} Let $\chi,\rho\in \cC^\infty(\bR,\bR)$ be convex increasing 
functions to be specified later. We use here the {\it hermitian} metric
$$\eqalign{
\alpha&=\ii\Theta(E_\chi)+\exp(-\rho\circ\psi)\,\omega\cr
      &=\ii\Theta(E)+\ii d'd''(\chi\circ\psi)+\exp(-\rho\circ\psi)\,\omega.
\cr}$$
Although $\omega$ is K\"ahler, the metric $\alpha$ is not so.
Denote by $\gamma_j^{\chi,\omega}$ (resp.\ $\gamma_j^{\chi,\alpha}$),
$1\le j\le n$, the eigenvalues of $\ii\Theta(E_\chi)$ with respect to 
$\omega$ (resp.\ $\alpha$), rearranged in increasing order. The minimax
principle implies $\gamma_j^{\chi,\omega}\ge\gamma_j^{0,\omega}$, and the
hypothesis yields $0<\gamma_s^{0,\omega}\le\gamma_{s+1}^{0,\omega}\le
\ldots\le\gamma_n^{0,\omega}$ on $X$. By means of a diagonalization of
$\ii\Theta(E_\chi)$ with respect to $\omega$, we find
$$1\ge\gamma_j^{\chi,\alpha}={\gamma_j^{\chi,\omega}\over
\gamma_j^{\chi,\omega}+\exp(-\rho\circ\psi)}\ge{\gamma_j^{0,\omega}\over
\gamma_j^{0,\omega}+\exp(-\rho\circ\psi)}.$$
Let $\varepsilon>0$ be small. Select $\rho$ such that
$\exp(-\rho\circ\psi(x))\le\varepsilon\gamma_s^{0,\omega}(x)$ at every
point. Then for $j\ge s$ we get
$$\gamma_j^{\chi,\alpha}\ge{\gamma_j^{0,\omega}\over
\gamma_j^{0,\omega}+\varepsilon\gamma_j^{0,\omega}}={1\over 1+\varepsilon}
\ge 1-\varepsilon,$$
and Th.~VI-8.3 implies
$$\eqalign{
\langle\big[\ii\Theta(E_\chi),\Lambda_\alpha\big]u,u\rangle_\alpha&\ge
\big(\gamma_1^{\chi,\alpha}+\cdots+\gamma_p^{\chi,\alpha}-
\gamma_{q+1}^{\chi,\alpha}-\ldots-\gamma_n^{\chi,\alpha}\big)|u|^2\cr
&\ge\big((p-s+1)(1-\varepsilon)-(n-q)\big)|u|^2\cr
&\ge\big(1-(p-s+1)\varepsilon\big)|u|^2.\cr}$$
It remains however to control the torsion term $T_\alpha$. As $\omega$
is K\"ahler, trivial computations yield
$$\eqalign{
d'\alpha&=-\rho'\circ\psi~\exp(-\rho\circ\psi)~d'\psi\wedge\omega,\cr
d'd''\alpha&=\exp(-\rho\circ\psi)~\big[\big((\rho'\circ\psi)^2-
\rho''\circ\psi\big)d'\psi\wedge d''\psi-\rho'\circ\psi~d'd''\psi\big]
\wedge\omega.\cr}$$
Since
$$\alpha\ge i(\chi'\circ\psi~d'd''\psi+\chi''\circ\psi~d'\psi\wedge d''\psi)
+\exp(-\rho\circ\psi)\omega,$$
we get the upper bounds
$$\eqalign{
|d'\alpha|_\alpha&\le\rho'\circ\psi~|d'\psi|_\alpha~|\exp(-\rho\circ\psi)
\omega|_\alpha\le\rho'\circ\psi~(\chi''\circ\psi)^{-{1\over 2}}\cr
|d'd''\alpha|_\alpha&\le{(\rho'\circ\psi)^2+\rho''\circ\psi\over\chi''\circ\psi}
+{\rho'\circ\psi\over\chi'\circ\psi}.\cr}$$
It is then clear that we can choose $\chi$ growing sufficiently fast
in order that $|T_\alpha|_\alpha\le\varepsilon$. If $\varepsilon$ is chosen
sufficiently small, we get $A_{E_\chi,\alpha}\ge{1\over 2}\,\Id$, 
and the conclusion is obtained in the same way as for Th.~5.6.\qed
\endproof

\section{6}{H\"ormander's Estimates for non Complete K\"ahler Metrics}
Our aim here is to derive also estimates for a non complete K\"ahler
metric, for example the standard metric of $\bC^n$ on a bounded domain
$\Omega\subset\!\subset\bC^n$. A result of this type can be obtained
in the situation described at the end of Remark~4.8. The underlying
idea is due to \cite{H\"ormander 1966}, although we do not apply his so
called ``three weights" technique, but use instead an approximation of
the given metric $\omega$ by complete K\"ahler metrics.

\claim{(6.1) Theorem} Let $(X,\wh\omega)$ be a complete K\"ahler manifold,
$\omega$ another K\"ahler metric, possibly non complete, and $E\longrightarrow X$ 
a $m$-semi-positive vector bundle. Let $g\in L^2_{n,q}(X,E)$ be such that 
$D''g=0$ and
$$\int_X\langle A_q^{-1}g,g\rangle\,dV<+\infty$$
with respect to $\omega$, where $A_q$ stands for the operator
$\ii\Theta(E)\wedge\Lambda$ in bidegree $(n,q)$ and $q\ge 1$, $m\ge\min\{n-q+1,r\}$. 
Then there exists $f\in L^2_{n,q-1}(X,E)$ such that $D''f=g$ and
$$\|f\|^2\le\int_X\langle A_q^{-1}g,g\rangle\,dV.$$
\endclaim

\proof{Proof} For every $\varepsilon>0$, the K\"ahler metric
$$\omega_\varepsilon=\omega+\varepsilon\wh\omega$$
is complete. The idea of the proof is to apply the $L^2$ estimates to 
$\omega_\varepsilon$ and to let $\varepsilon$ tend to zero. Let us put an
index $\varepsilon$ to all objects depending on $\omega_\varepsilon$. 
It follows from Lemma~6.3 below that
$$|u|_\varepsilon^2\,dV_\varepsilon\le|u|^2\,dV,~~~~
\langle A_{q,\varepsilon}^{-1}u,u\rangle_\varepsilon\,dV_\varepsilon\le
\langle A_q^{-1}u,u\rangle\,dV\leqno(6.2)$$
for every $u\in\Lambda^{n,q}T^\star_X\otimes E$. If these estimates are 
taken for granted, Th.~4.5 applied to $\omega_\varepsilon$
yields a section $f_\varepsilon\in L^2_{n,q-1}(X,E)$
such that $D''f_\varepsilon=g$ and
$$\int_X|f_\varepsilon|_\varepsilon^2\,dV_\varepsilon\le
\int_X\langle A_{q,\varepsilon}^{-1}g,g\rangle_\varepsilon\,dV_\varepsilon\le
\int_X\langle A_q^{-1}g,g\rangle\,dV.$$
This implies that the family $(f_\varepsilon)$ is bounded in $L^2$ norm
on every compact subset of $X$. We can thus find a weakly convergent
subsequence $(f_{\varepsilon_\nu})$ in $L^2_\loc$. The weak limit $f$
is the solution we are looking for.\qed
\endproof

\claim{(6.3) Lemma} Let $\omega$, $\gamma$ be hermitian metrics on
$X$ such that $\gamma\ge\omega$. For every $u\in\Lambda^{n,q}
T^\star_X\otimes E$, $q\ge 1$, we have
$$|u|_\gamma^2\,dV_\gamma\le|u|^2\,dV,~~~~
\langle A_{q,\gamma}^{-1}u,u\rangle_\gamma\,dV_\gamma\le
\langle A_q^{-1}u,u\rangle\,dV$$
where an index $\gamma$ means that the corresponding term is computed
in terms of $\gamma$ instead of $\omega$.
\endclaim

\proof{Proof} Let $x_0\in X$ be a given point and
$(z_1,\ldots,z_n)$ coordinates such that
$$\omega=\ii\sum_{1\le j\le n}dz_j\wedge d\ol z_j,~~~~
  \gamma=\ii\sum_{1\le j\le n}\gamma_j\,dz_j\wedge d\ol z_j~~~
\hbox{\rm at}~~x_0,$$
where $\gamma_1\le\ldots\le\gamma_n$ are the eigenvalues of 
$\gamma$ with respect to $\omega$ (thus $\gamma_j\ge 1$).
We have $|dz_j|_\gamma^2=\gamma_j^{-1}$ and $|dz_K|_\gamma^2=
\gamma_K^{-1}$ for any multi-index $K$, with the notation
$\gamma_K=\prod_{j\in K}\gamma_j$. 
For every $\,u=\sum u_{K,\lambda}dz_1\wedge\ldots\wedge dz_n\wedge d\ol z_K
\otimes e_\lambda$, $|K|=q$, $1\le\lambda\le r$, the computations of 
\S~VII-7 yield
$$\leqalignno{
|u|_\gamma^2&=\sum_{K,\lambda}~(\gamma_1\ldots\gamma_n)^{-1}
\gamma_K^{-1}\,|u_{K,\lambda}|^2,~~~~
dV_\gamma=\gamma_1\ldots\gamma_n\,dV,\cr
|u|_\gamma^2\,dV_\gamma&=\sum_{K,\lambda}\gamma_K^{-1}\,
|u_{K,\lambda}|^2\,dV\le|u|^2\,dV,\cr
\Lambda_\gamma u&=\sum_{|I|=q-1}\sum_{j,\lambda}\ii(-1)^{n+j-1}\gamma_j^{-1}\,
u_{jI,\lambda}\,(\wh{dz_j})\wedge d\ol z_I
\otimes e_\lambda,\cr}$$
where $(\wh{dz_j})$ means $dz_1\wedge\ldots\wh{dz_j}\ldots \wedge dz_n$,
$$\leqalignno{
A_{q,\gamma}u&=\sum_{|I|=q-1}\sum_{j,k,\lambda,\mu}\gamma_j^{-1}\,
c_{jk\lambda\mu}\,u_{jI,\lambda}\,dz_1\wedge\ldots\wedge dz_n\wedge
d\ol z_{kI}\otimes e_\mu,\cr
\llap{\hbox{$\langle A_{q,\gamma}u,u\rangle_\gamma$}}&=
(\gamma_1\ldots\gamma_n)^{-1}\sum_{|I|=q-1}\gamma_I^{-1}
\sum_{j,k,\lambda,\mu}\gamma_j^{-1}\gamma_k^{-1}c_{jk\lambda\mu}\,
u_{jI,\lambda}\ol u_{kI,\mu}\cr
&\ge(\gamma_1\ldots\gamma_n)^{-1}\sum_{|I|=q-1}\gamma_I^{-2}
\sum_{j,k,\lambda,\mu}\gamma_j^{-1}\gamma_k^{-1}c_{jk\lambda\mu}\,
u_{jI,\lambda}\ol u_{kI,\mu}\cr
&=\gamma_1\ldots\gamma_n\,\langle A_qS_\gamma u,S_\gamma u\rangle\cr}$$
where $S_\gamma$ is the operator defined by
$$S_\gamma u=\sum_K~(\gamma_1\ldots\gamma_n\gamma_K)^{-1}\,
u_{K,\lambda}\,dz_1\wedge\ldots\wedge dz_n\wedge d\ol z_K\otimes e_\lambda.$$
We get therefore
$$\eqalign{
|\langle u,v\rangle_\gamma|^2=|\langle u,S_\gamma v\rangle|^2&\le
\langle A_q^{-1}u,u\rangle\langle A_q S_\gamma v,S_\gamma v\rangle\cr
&\le(\gamma_1\ldots\gamma_n)^{-1}\langle A_q^{-1}u,u\rangle
\langle A_{q,\gamma}v,v\rangle_\gamma,\cr}$$
and the choice $v=A_{q,\gamma}^{-1}u$ implies
$$\langle A_{q,\gamma}^{-1}u,u\rangle_\gamma\le
(\gamma_1\ldots\gamma_n)^{-1}\,\langle A_q^{-1}u,u\rangle\,;$$
this relation is equivalent to the last one in the lemma.\qed
\endproof

An important special case is that of a semi-positive line bundle $E$. If we
let $0\le\lambda_1(x)\le\ldots\le\lambda_n(x)$ be the eigenvalues of $\ii\Theta(E)_x$ 
with respect to $\omega_x$ for all $x\in X$, formula VI-8.3 implies
$$\leqalignno{
\langle A_qu,u\rangle&\ge(\lambda_1+\cdots+\lambda_q)|u|^2,\cr
\int_X\langle A_q^{-1}g,g\rangle\,dV&\le\int_X~{1\over\lambda_1+\cdots+
\lambda_q}\,|g|^2\,dV.&(6.4)\cr}$$
A typical situation where these estimates can be applied is the case when
$E$ is the trivial line bundle $X\times\bC$ with metric given by a weight 
$e^{-\varphi}$.
One can assume for example that $\varphi$ is plurisubharmonic and that
$\ii d'd''\varphi$ has at least $n-q+1$ positive eigenvalues at every point,
i.e.\ $\lambda_q>0$ on $X$. This situation leads to very important 
$L^2$ estimates, which are precisely those given by \cite{H\"ormander~1965,
1966}. We state here a slightly more general result.

\claim{(6.5) Theorem} Let $(X,\omega)$ be a weakly pseudoconvex K\"ahler
manifold, $E$ a hermitian line bundle on $X$,
$\varphi\in \cC^\infty(X,\bR)$ a weight function such that the eigenvalues
$\lambda_1\le\ldots\le\lambda_n$ of $\ii\Theta(E)+\ii d'd''\varphi$ are $\ge0$.
Then for every form $g$ of type $(n,q)$, $q\ge 1$, with $L^2_\loc$ 
(resp.\ $\cC^\infty$) coefficients such that $D''g=0$ and
$$\int_X~{1\over\lambda_1+\cdots+\lambda_q}\,|g|^2\,e^{-\varphi}\,dV<+\infty,$$
we can find a $L^2_\loc$ (resp.\ $\cC^\infty$) form $f$ of type $(n,q-1)$
such that $D''f=g$ and
$$\int_X~|f|^2\,e^{-\varphi}\,dV\le\int_X~{1\over\lambda_1+\cdots+
\lambda_q}\,|g|^2\,e^{-\varphi}\,dV.$$
\endclaim

\proof{Proof} Apply the general estimates to the bundle $E_\varphi$ deduced
from $E$ by multiplication of the metric by $e^{-\varphi}\,;$ we have
$\ii\Theta(E_\varphi)=\ii\Theta(E)+\ii d'd''\varphi$.
It is not necessary here to assume in addition that $g\in L^2_{n,q}
(X,E_\varphi)$. In fact, $g$ is in $L^2_\loc$ and we can exhaust X by
the relatively compact weakly pseudoconvex domains
$$X_c=\big\{x\in X~;~\psi(x)<c\big\}$$
where $\psi\in \cC^\infty(X,\bR)$ is a plurisubharmonic exhaustion function
(note that \hbox{$-\log(c-\psi)$} is also such a function on $X_c$). We
get therefore solutions $f_c$ on $X_c$ with uniform $L^2$ bounds; any
weak limit $f$ gives the desired solution.\qed
\endproof

If estimates for $(p,q)$-forms instead of $(n,q)$-forms are needed, one
can invoke the isomorphism $\Lambda^pT^\star_X\simeq\Lambda^{n-p}T_X\otimes
\Lambda^nT^\star_X$ (obtained through contraction of $n$-forms by
$(n-p)$-vectors) to get
$$\Lambda^{p,q}T^\star_X\otimes E\simeq\Lambda^{n,q}T^\star_X\otimes F,~~~~
F=E\otimes\Lambda^{n-p}T_X.$$
Let us look more carefully to the case $p=0$. The $(1,1)$-curvature form of 
$\Lambda^nT_X$ with respect to a hermitian metric $\omega$ on $T_X$ is called 
the {\it Ricci curvature} of $\omega$. We denote:

\claim{(6.6) Definition} $\Ric(\omega)=\ii\Theta(\Lambda^n T_X)=
\ii\,\Tr\,\Theta(T_X)$.
\endclaim

For any local coordinate system $(z_1,\ldots,z_n)$, the holomorphic
$n$-form $dz_1\wedge\ldots\wedge dz_n$ is a section of $\Lambda^n
T^\star_X$, hence Formula V-13.3 implies
$$\Ric(\omega)=\ii d'd''\log|dz_1\wedge\ldots\wedge dz_n|_\omega^2=
-\ii d'd''\log\det(\omega_{jk}).\leqno(6.7)$$
The estimates of Th.~6.5 can therefore be applied to any $(0,q)$-form
$g$, but $\lambda_1\le\ldots\le\lambda_n$ must be replaced by the
eigenvalues of the $(1,1)$-form
$$\ii\Theta(E)+\Ric(\omega)+\ii d'd''\varphi~~~~{\rm(supposed~~}\ge0).\leqno(6.8)$$

We consider now domains $\Omega\subset\bC^n$ equipped
with the euclidean metric of $\bC^n$, and the trivial bundle
$E=\Omega\times\bC$.
The following result is especially convenient because it requires only
weak plurisubharmonicity and avoids to compute the curvature eigenvalues.

\claim{(6.9) Theorem} Let $\Omega\subset\bC^n$ be a weakly pseudoconvex
open subset and $\varphi$ an upper semi-continuous plurisubharmonic
function on $\Omega$. For every $\varepsilon\in{}]0,1]$ and every
$g\in L^2_{p,q}(\Omega,\loc)$ such that $d''g=0$ and
$$\int_\Omega~\big(1+|z|^2\big)|g|^2\,e^{-\varphi}dV<+\infty,$$
we can find a $L^2_\loc$ form $f$ of type $(p,q-1)$ such that $d''f=g$ and
$$\int_\Omega~\big(1+|z|^2\big)^{-\varepsilon}\,|f|^2\,e^{-\varphi}\,dV\le
{4\over q\varepsilon^2}
\int_\Omega~\big(1+|z|^2\big)|g|^2\,e^{-\varphi}\,dV<+\infty.$$
Moreover $f$ can be chosen smooth if $g$ and $\varphi$ are smooth.
\endclaim

\proof{Proof} Since $\Lambda^pT\Omega$ is a trivial bundle with
trivial metric, the proof is immediately reduced to the case $p=0$
(or equivalently $p=n$). Let us first suppose that $\varphi$ is smooth.
We replace $\varphi$ by $\Phi=\varphi+\tau$ where
$$\tau(z)=\log\big(1+(1+|z|^2)^\varepsilon\big).$$
\endproof

\claim{(6.10) Lemma} The smallest eigenvalue $\lambda_1(z)$ of
$\ii d'd''\tau(z)$ satisfies
$$\lambda_1(z)\ge{\varepsilon^2\over 2(1+|z|^2)\big(1+(1+|z|^2)^\varepsilon
\big)}.$$
\endclaim

In fact a brute force computation of the complex hessian $H\tau_z(\xi)$
and the Cauchy-Schwarz inequality yield
$$\eqalignno{
&\qquad\qquad\qquad\qquad H\tau_z(\xi)=\cr
&={\varepsilon(1{+}|z|^2)^{\varepsilon-1}|\xi|^2\over
1+(1{+}|z|^2)^\varepsilon}{+}{\varepsilon(\varepsilon-1)
(1{+}|z|^2)^{\varepsilon-2}|\langle\xi,z\rangle|^2\over 1+(1{+}|z|^2)^\varepsilon}
{-}{\varepsilon^2(1{+}|z|^2)^{2\varepsilon-2}|\langle\xi,z\rangle|^2\over
\big(1+(1{+}|z|^2)^\varepsilon\big)^2}\cr
&\ge\varepsilon\bigg({(1+|z|^2)^{\varepsilon-1}\over
1+(1+|z|^2)^\varepsilon}-{(1-\varepsilon)
(1+|z|^2)^{\varepsilon-2}|z|^2\over 1+(1+|z|^2)^\varepsilon}
-{\varepsilon(1+|z|^2)^{2\varepsilon-2}|z|^2\over
\big(1+(1+|z|^2)^\varepsilon\big)^2}\bigg)|\xi|^2\cr
&=\varepsilon\,{1+\varepsilon|z|^2+(1+|z|^2)^\varepsilon\over
(1+|z|^2)^{2-\varepsilon}\big(1+(1+|z|^2)^\varepsilon\big)^2}\,|\xi|^2
\ge{\varepsilon^2|\xi|^2\over(1+|z|^2)^{1-\varepsilon}
\big(1+(1+|z|^2)^\varepsilon\big)^2}\cr
&\ge{\varepsilon^2\over 2(1+|z|^2)\big(1+(1+|z|^2)^\varepsilon\big)}
\,|\xi|^2.&\square\cr}$$
\medskip

The Lemma~implies $e^{-\tau}/\lambda_1\le 2(1+|z|^2)/\varepsilon^2$,
thus Cor.~6.5 provides an $f$ such that
$$\int_\Omega~\big(1+(1+|z|^2)^\varepsilon\big)^{-1}\,|f|^2\,e^{-\varphi}\,dV
\le{2\over q\varepsilon^2}
\int_\Omega~\big(1+|z|^2\big)|g|^2\,e^{-\varphi}\,dV<+\infty,$$
and the required estimate follows. If $\varphi$ is not smooth, apply
the result to a sequence of regularized weights $\rho_\varepsilon\star\varphi\ge
\varphi$ on an increasing sequence of domains $\Omega_c\subset\!\subset\Omega$,
and extract a weakly convergent subsequence of solutions.\qed

\section{7}{Extension of Holomorphic Functions from Subvarieties}
The existence theorems for solutions of the $d''$ operator easily lead to
an extension theorem for sections of a holomorphic line bundle defined
in a neighborhood of an analytic subset. The following result
\cite{Demailly~1982} is  an improvement and a generalization of Jennane's
extension theorem \cite{Jennane 1976}.

\claim{(7.1) Theorem} Let $(X,\omega)$ be a weakly pseudoconvex K\"ahler
manifold, $L$ a hermitian line bundle and $E$ a hermitian vector bundle
over $X$.  Let $Y$ be an analytic subset of $X$ such that
$Y=\sigma^{-1}(0)$ for some section $\sigma$ of $E$, and $p$ the
maximal codimension of the irreducible components of $Y$.  Let $f$ be a
holomorphic section of $K_X\otimes L$ defined in the open set $U\supset Y$
of points \hbox{$x\in X$} such that $|\sigma(x)|<1$.
If $\int_U|f|^2dV<+\infty$ and if the curvature form of $L$ satisfies
$$\ii\Theta(L)\ge\Big({p\over|\sigma|^2}+{\varepsilon\over 1+|\sigma|^2}\Big)
\{\ii\Theta(E)\sigma,\sigma\}$$
for some $\varepsilon>0$, there is a section $F\in H^0(X,K_X\otimes L)$ 
such that \hbox{$F_{\restriction Y}=f_{\restriction Y}$} and
$$\int_X{|F|^2\over(1+|\sigma|^2)^{p+\varepsilon}}\,dV\le
\Big(1+{(p+1)^2\over\varepsilon}\Big)\int_U|f|^2\,dV.$$
\endclaim

The proof will involve a weight with logarithmic singularities along $Y$.
We must therefore apply the existence theorem over $X\ssm Y$.
This requires to know whether $X\ssm Y$ has a complete K\"ahler metric.

\claim{(7.2) Lemma} Let $(X,\omega)$ be a K\"ahler manifold, and
$Y=\sigma^{-1}(0)$ an analytic subset defined by a section of a
hermitian vector bundle $E\to X$.  If $X$ is weakly pseudoconvex and
exhausted by $X_c=\{x\in X~;~\psi(x)<c\}$, then $X_c\ssm Y$ has a
complete K\"ahler metric for all $c\in\bR$.  The same conclusion
holds for $X\ssm Y$ if $(X,\omega)$ is complete and if for some
constant $C\ge 0$ we have
$\Theta_E\le_\Grif C\,\omega\otimes\langle~,~\rangle_E$ on $X$.
\endclaim

\proof{Proof} Set $\tau=\log|\sigma|^2$. Then $d'\tau=\{D'\sigma,\sigma\}/
|\sigma|^2$ and $D''D'\sigma=D^2\sigma=\Theta(E)\sigma$, thus
$$\ii d'd''\tau=\ii{\{D'\sigma,D'\sigma\}\over|\sigma|^2}-
\ii{\{D'\sigma,\sigma\}\wedge\{\sigma,D'\sigma\}\over|\sigma|^4}-
{\{\ii\Theta(E)\sigma,\sigma\}\over|\sigma|^2}.$$
For every $\xi\in T_X$, we find therefore
$$\eqalign{
H\tau(\xi)&={|\sigma|^2\,|D'\sigma\cdot \xi|^2-|\langle D'\sigma\cdot \xi,
\sigma\rangle|^2\over|\sigma|^4}-{\Theta_E(\xi\otimes\sigma,\xi\otimes
\sigma)\over|\sigma|^2}\cr
&\ge -{\Theta_E(\xi\otimes \sigma,\xi\otimes\sigma)\over|\sigma|^2}\cr}$$
by the Cauchy-Schwarz inequality. If $C$ is a bound for the coefficients of
$\Theta_E$ on the compact
subset $\ol X_c$, we get $\ii d'd''\tau\ge -C\omega$ on
$X_c$. Let $\chi\in \cC^\infty(\bR,\bR)$ be a convex increasing function. We set
$$\wh\omega=\omega+\ii d'd''(\chi\circ\tau).$$
Formula 5.3 shows that $\wh\omega$ is positive definite if
$\chi'\le 1/2C$ and that $\wh\omega$ is complete near $Y=\tau^{-1}(-\infty)$
as soon as
$$\int_{-\infty}^0\sqrt{\chi''(t)}\,dt=+\infty.$$
One can choose for example $\chi$ such that
$\chi(t)={1\over 5C}(t-\log|t|)$ for $t\le -1$. In order to obtain a
complete K\"ahler metric on $X_c\ssm Y$, we need also that the metric 
be complete near $\partial X_c$. Such a metric is given by
$$\eqalign{
\wt\omega&=\wh\omega+\ii d'd''\log(c-\psi)^{-1}=\wh\omega+{\ii d'd''\psi\over
c-\psi}+{\ii d'\psi\wedge d''\psi\over(c-\psi)^2}\cr
&\ge \ii d'\log(c-\psi)^{-1}\wedge d''\log(c-\psi)^{-1}~;\cr}$$
$\wt\omega$ is complete on $X_c\ssm\Omega$ because $\log(c-\psi)^{-1}$
tends to $+\infty$ on $\partial X_c$.\qed
\endproof

\proof{Proof of Theorem 7.1.} When we replace $\sigma$ by 
$(1+\eta)\sigma$ for some small $\eta>0$ and let $\eta$ tend to $0$, we see 
that we can assume $f$ defined in a neighborhood of $\ol U$. Let $h$ be the 
continuous section of $L$ such that $h=(1-|\sigma|^{p+1})f$ on 
$U=\{|\sigma|<1\}$ and $h=0$ on $X\ssm U$. 
We have $h_{\restriction Y}=f_{\restriction Y}$ and
$$d''h=-{p+1\over 2}\,|\sigma|^{p-1}\,\{\sigma,D'\sigma\}\,f~~~~{\rm on}~~U,
~~~~d''h=0~~~~{\rm on}~~X\ssm U.$$
We consider $g=d''h$ as a $(n,1)$-form with values in the hermitian line
bundle $L_\varphi=L$, endowed with the weight $e^{-\varphi}$ given by
$$\varphi=p\log|\sigma|^2+\varepsilon\log(1+|\sigma|^2).$$
Notice that $\varphi$ is singular along $Y$.
The Cauchy-Schwarz inequality implies $\ii\{D'\sigma,\sigma\}
\wedge\{\sigma,D'\sigma\}\le\ii\{D'\sigma,D'\sigma\}$ as in Lemma~7.2,
and we find
$$\eqalign{
\ii d'd''\log(1+|\sigma|^2)&={(1+|\sigma|^2)\ii\{D'\sigma,D'\sigma\}-
\ii\{D'\sigma,\sigma\}\wedge\{\sigma,D'\sigma\}\over(1+|\sigma|^2)^2}\cr
&\quad{}-{\{\ii\Theta(E)\sigma,\sigma\}\over1+|\sigma|^2}
\ge{\ii\{D'\sigma,D'\sigma\}\over 
(1+|\sigma|^2)^2}-{\{\ii\Theta(E)\sigma,\sigma\}\over1+|\sigma|^2}.\cr}$$
The inequality $\ii d'd''\log|\sigma|^2\ge-\{\ii\Theta(E)\sigma,\sigma\}/|\sigma|^2$ 
obtained in Lemma~7.2 and the above one imply
$$\eqalign{\ii\Theta(L_\varphi)
&=\ii\Theta(L)+p\,\ii d'd''\log|\sigma|^2+\varepsilon\,\ii d'd''\log(1+|\sigma|^2)\cr
&\ge \ii\Theta(L)-\Big({p\over|\sigma|^2}+{\varepsilon\over 1+|\sigma|^2}\Big)\{\ii\Theta(E)
\sigma,\sigma\}+\varepsilon{\ii\{D'\sigma,D'\sigma\}\over (1+|\sigma|^2 )^2}\cr
&\ge\varepsilon{\ii\,\{D'\sigma,\sigma\}\wedge\{\sigma,D'\sigma\}\over|\sigma|^2
\,(1+|\sigma|^2 )^2},\cr}$$
thanks to the hypothesis on the curvature of $L$ and the Cauchy-Schwarz
inequality. Set $\xi=(p+1)/2\,|\sigma|^{p-1}\{D'\sigma,\sigma\}
=\sum\xi_j\,dz_j$ in an $\omega$-orthonormal basis $\partial/\partial z_j$,
and let $\wh\xi=\sum\xi_j\partial/\partial\ol z_j$ be the dual $(0,1)$-vector 
field. For every $(n,1)$-form $v$ with values in $L_\varphi$, we find
$$\eqalign{
\big|\langle d''h,v\rangle\big|&=\big|\langle\ol\xi\wedge f,v\rangle\big|=
\big|\langle f,\wh\xi\ort v\rangle\big|\le|f|\,|\wh\xi\ort v|,\cr
\wh\xi\ort v&=\sum-\ii\xi_j\,dz_j\wedge\Lambda v=-\ii\xi\wedge\Lambda v,\cr
|\langle d''h,v\rangle|^2&\le|f|^2\,|\wh\xi\ort v|^2=
|f|^2\langle-\ii\xi\wedge\Lambda v,\wh\xi\ort v\rangle\cr
&=|f|^2\langle-\ii\ol\xi\wedge\xi\wedge\Lambda v,v\rangle=
|f|^2\langle[\ii\xi\wedge\ol\xi,\Lambda]v,v\rangle\cr
&\le{(p+1)^2\over 4\varepsilon}\,|\sigma|^{2p}\,(1+|\sigma|^2)^2\,|f|^2\,
\langle[\ii\Theta(L_\varphi),\Lambda]v,v\rangle.}$$
Thus, in the notations of Th.~6.1, the form $g=d''h$ satisfies
$$\langle A_1^{-1}g,g\rangle\le
{(p+1)^2\over4\varepsilon}\,|\sigma|^{2p}(1+|\sigma|^2)^2\,|f|^2\le
{(p+1)^2\over\varepsilon}\,|f|^2\,e^\varphi,$$
where the last equality results from the fact that $(1+|\sigma|^2)^2\le 4$
on the support of $g$. Lemma 7.2 shows that the existence theorem
6.1 can be applied on each set $X_c\ssm Y$. Letting $c$
tend to infinity, we infer the existence of a 
$(n,0)$-form $u$ with values in $L$ such that $d''u=g$ on 
$X\ssm Y$ and
$$\eqalign{
\int_{X\ssm Y}|u|^2\,e^{-\varphi}\,dV&\le
\int_{X\ssm Y}\langle A_1^{-1}g,g\rangle e^{-\varphi},~~~~{\rm thus}\cr
\int_{X\ssm Y}{|u|^2\over|\sigma|^{2p}(1+|\sigma|^2)^\varepsilon}\,dV
&\le{(p+1)^2\over\varepsilon}\int_U|f|^2\,dV.\cr}$$
This estimate implies in particular that $u$ is locally $L^2$ near $Y$. 
As $g$ is conti\-nuous over $X$, Lemma~7.3 below shows that the equality
\hbox{$d''u=g=d''h$} extends to $X$, thus $F=h-u$ is holomorphic everywhere.
Hence \hbox{$u=h-F$} is continuous on $X$.
As $|\sigma(x)|\le C\,d(x,Y)$ in a neighborhood of
every point of $Y$, we see that $|\sigma|^{-2p}$ is non integrable
at every point $x_0\in Y_{\rm reg}$, because $\codim Y\le p$. It follows 
that $u=0$ on $Y$, so
$$F_{\restriction Y}=h_{\restriction Y}=f_{\restriction Y}.$$
The final $L^2$-estimate of Th.~7.1 follows from the inequality
$$|F|^2=|h-u|^2\le(1+|\sigma|^{-2p})\,|u|^2+(1+|\sigma|^{2p})\,|f|^2$$
which implies
$${|F|^2\over(1+|\sigma|^2)^p}\le{|u|^2\over|\sigma|^{2p}}+|f|^2.
\eqno{\square}$$
\endproof

\claim{(7.3) Lemma} Let $\Omega$ be an open subset of $\bC^n$ and
$Y$ an analytic subset of $\Omega$. Assume that $v$ is a $(p,q-1)$-form
with $L^2_\loc$ coefficients and $w$ a $(p,q)$-form with 
$L^1_\loc$ coefficients such that $d''v=w$ on $\Omega\ssm Y$
$($in the sense of distribution theory$)$. Then $d''v=w$ on $\Omega$.
\endclaim

\proof{Proof} An induction on the dimension of $Y$ shows that it is
sufficient to prove the result in a neighborhood of a regular point $a\in Y$.
By using a local analytic isomorphism, the proof is reduced to the case
where $Y$ is contained in the hyperplane $z_1=0$, with $a=0$. Let
$\lambda\in \cC^\infty(\bR,\bR)$ be a function such that $\lambda(t)=0$ for
$t\le{1\over 2}$ and $\lambda(t)=1$ for $t\ge 1$. We must show that
$$\int_\Omega w\wedge\alpha=(-1)^{p+q}\int_\Omega v\wedge d''\alpha
\leqno(7.4)$$
for all $\alpha\in\cD_{n-p,n-q}(\Omega)$. Set $\lambda_\varepsilon(z)=
\lambda(|z_1|/\varepsilon)$ and replace $\alpha$ in the integral by
$\lambda_\varepsilon\alpha$. Then $\lambda_\varepsilon\alpha\in
\cD_{n-p,n-q}(\Omega\ssm Y)$ and the hypotheses imply
$$\int_\Omega w\wedge\lambda_\varepsilon\alpha=(-1)^{p+q}
\int_\Omega v\wedge d''(\lambda_\varepsilon\alpha)=(-1)^{p+q}
\int_\Omega v\wedge (d''\lambda_\varepsilon\wedge\alpha+\lambda_\varepsilon 
d''\alpha).$$
As $w$ and $v$ have $L^1_\loc$ coefficients on $\Omega$, the integrals
of $w\wedge\lambda_\varepsilon\alpha$ and $v\wedge\lambda_\varepsilon 
d''\alpha$ converge respectively to the integrals of $w\wedge\alpha$
and $v\wedge d''\alpha$ as $\varepsilon$ tends to $0$. The remaining
term can be estimated by means of the Cauchy-Schwarz inequality:
$$\Big|\int_\Omega v\wedge d''\lambda_\varepsilon\wedge\alpha\Big|^2\le
\int_{|z_1|\le\varepsilon}|v\wedge\alpha|^2\,dV.~
\int_{\Supp\alpha}|d''\lambda_\varepsilon|^2\,dV\,;$$
as $v\in L^2_\loc(\Omega)$, the integral
$\int_{|z_1|\le\varepsilon}|v\wedge\alpha|^2\,dV$ converges to $0$ with 
$\varepsilon$, whereas
$$\int_{\Supp\alpha}|d''\lambda_\varepsilon|^2\,dV\le
{C\over\varepsilon^2}\,{\rm Vol}\big(\Supp\,\alpha
\cap\{|z_1|\le\varepsilon\}\big)\le C'.$$
Equality (7.4) follows when $\varepsilon$ tends to $0$.\qed
\endproof

\claim{(7.5) Corollary} Let $\Omega\subset\bC^n$ be a weakly pseudoconvex
domain and let $\varphi$, $\psi$ be plurisubharmonic functions on $\Omega$,
where $\psi$ is supposed to be finite and continuous. Let $\sigma=(\sigma_1,\ldots,
\sigma_r)$ be a family of holomorphic functions on $\Omega$, let
\hbox{$Y=\sigma^{-1}(0)$}, $p={}$  maximal codimension of $Y$ and set
\smallskip
\item{\rm a)} $U=\{z\in\Omega\,;\,|\sigma(z)|^2<e^{-\psi(z)}\}$,~~~~resp.\
\smallskip
\item{\rm b)} $U'=\{z\in\Omega\,;\,|\sigma(z)|^2<e^{\psi(z)}\}$.
\smallskip\noindent
For every $\varepsilon>0$ and every holomorphic function $f$ on $U$,
there exists a holomorphic function $F$ on $\Omega$ such that
$F_{\restriction Y}=f_{\restriction Y}$ and
$$\leqalignno{
\int_\Omega {|F|^2\,e^{-\varphi+p\psi}\over
(1+|\sigma|^2e^{\psi})^{p+\varepsilon}}\,dV&\le
\Big(1+{(p+1)^2\over\varepsilon}\Big)\int_U|f|^2\,e^{-\varphi+p\psi}\,dV,
~~~~{\it resp.}&{\rm a)}\cr
\int_\Omega {|F|^2\,e^{-\varphi}\over
(e^\psi+|\sigma|^2)^{p+\varepsilon}}\,dV&\le
\Big(1+{(p+1)^2\over\varepsilon}\Big)\int_U|f|^2\,e^{-\varphi-(p+
\varepsilon)\psi}\,dV.&{\rm b)}\cr}$$
\endclaim

\proof{Proof} After taking convolutions with smooth kernels on pseudoconvex
subdomains $\Omega_c\subset\!\subset\Omega$, we may assume $\varphi$,
$\psi$ smooth. In either case a) or~b), apply Th.~7.1 to
\medskip
\noindent a)~~$E=\Omega\times\bC^r$ with the weight $e^\psi$, 
$L=\Omega\times\bC$ with the weight $e^{-\varphi+p\psi}$, and 
$U=\{|\sigma|^2e^\psi<1\}$. Then
$$\ii\Theta(E)=-\ii d'd''\psi\otimes\Id_E\le 0,~~~~\ii\Theta(L)=\ii d'd''\varphi-
p\,\ii d'd''\psi\ge p\,\ii\Theta(E).$$
\smallskip
\noindent b)~~$E=\Omega\times\bC^r$ with the weight $e^{-\psi}$, 
$L=\Omega\times\bC$ with the weight $e^{-\varphi-(p+\varepsilon)\psi}$, and 
$U=\{|\sigma|^2e^{-\psi}<1\}$. Then
$$\ii\Theta(E)=\ii d'd''\psi\otimes\Id_E\ge 0,~~~~\ii\Theta(L)=\ii d'd''\varphi+
(p+\varepsilon)\,\ii d'd''\psi\ge(p+\varepsilon)\,\ii\Theta(E).$$ 
The condition on $\Theta(L)$ is satisfied in both cases and
$K_{\Omega}$ is trivial.\qed
\endproof

\claim{(7.6) H\"ormander-Bombieri-Skoda theorem} Let
$\Omega\subset\bC^n$ be a weakly pseudoconvex domain and $\varphi$ a
plurisubharmonic function on $\Omega$. For every $\varepsilon>0$ and
every point $z_0\in\Omega$ such that $e^{-\varphi}$ is integrable in a
neighborhood of $z_0$, there exists a holomorphic function $F$ on
$\Omega$ such that $F(z_0)=1$ and
$$\int_\Omega{|F(z)|^2\,e^{-\varphi(z)}\over(1+|z|^2)^{n+\varepsilon}}\,dV<
+\infty.$$
\endclaim

\cite{Bombieri~1970} originally stated the theorem with the exponent
$3n$ instead of $n+\varepsilon\,;$ the improved exponent
$n+\varepsilon$ is due to \cite{Skoda~1975}. The example
$\Omega=\bC^n$, $\varphi(z)=0$ shows that one cannot replace
$\varepsilon$ by $0$.

\proof{Proof} Apply Cor.~7.5 b) to $f\equiv 1$, $\sigma(z)=z-z_0$, $p=n$
and $\psi\equiv\log r^2$ where $U=B(z_0,r)$ is a ball such that
$\int_U e^{-\varphi}\,dV<+\infty$.\qed
\endproof

\claim{(7.7) Corollary} Let $\varphi$ be a plurisubharmonic function on a
complex ma\-ni\-fold $X$. Let $A$ be the set of points $z\in X$ such that
$e^{-\varphi}$ is not locally integrable in a neighborhood of $z$.
Then $A$ is an analytic subset of $X$.
\endclaim

\proof{Proof} Let $\Omega\subset X$ be an open coordinate patch
isomorphic to a ball of~$\bC^n$, with coordinates $(z_1,\ldots,z_n)$. 
Define $E\subset H^0(\Omega,\cO)$ to be the Hilbert space of holomorphic 
functions $f$ on $\Omega$ such that
$$\int_\Omega |f(z)|^2e^{-\varphi(z)}\,dV(z)<+\infty.$$
Then $A\cap\Omega=\bigcap_{f\in E}f^{-1}(0)$.
In fact, every $f$ in $E$ must obviously vanish on~$A\,;$ conversely,
if $z_0\notin A$, Th.~7.6 shows that there exists $f\in E$
such that $f(z_0)\ne 0$. By Th.~II-5.5, we conclude that $A$ is
analytic.\qed
\endproof

\section{8}{Applications to Hypersurface Singularities}
We first give some basic definitions and results concerning
multiplicities of divisors on a complex manifold.

\claim{(8.1) Proposition} Let $X$ be a complex manifold and 
$\Delta=\sum\lambda_j[Z_j]$ a divisor on $X$ with real coefficients
$\lambda_j\ge 0$. Let $x\in X$ and $f_j=0$, $1\le j\le N$, irreducible 
equations of $Z_j$ on a neighborhood $U$ of $x$.
\smallskip
\item{\rm a)}~~The multiplicity of $\Delta$ at $x$ is defined by
$$\mu(\Delta,x)=\sum\lambda_j\,{\rm ord}_x f_j.$$
\smallskip
\item{\rm b)} $\Delta$ is said to have normal crossings at a 
point $x\in\Supp\,\Delta$ if all hypersurfaces $Z_j$ containing $x$ are
smooth at $x$ and intersect transversally, i.e.\ if the linear forms $df_j$ 
defining the corresponding tangent spaces $T_xZ_j$ are linearly independent 
at $x$. The set ${\rm nnc}(\Delta)$ of non normal crossing points is an 
analytic subset of $X$.
\smallskip
\item{\rm c)} The non-integrability locus ${\rm nil}(\Delta)$ is defined
as the set of points $x\in X$ such that $\prod|f_j|^{-2\lambda_j}$
is non integrable near $x$. Then ${\rm nil}(\Delta)$ is an analytic subset
of $X$ and there are inclusions
$$\{x\in X\,;\,\mu(\Delta,x)\ge n\}\subset{\rm nil}(\Delta)\subset
\{x\in X\,;\,\mu(\Delta,x)\ge 1\}.$$
\item{}Moreover ${\rm nil}(\Delta)\subset{\rm nnc}(\Delta)$ if all
coefficients of $\Delta$ \hbox{satisfy $\lambda_j<1$}.
\endclaim

\proof{Proof} b) The set ${\rm nnc}(\Delta)\cap U$ is the union of the analytic
sets
$$f_{j_1}=\ldots=f_{j_p}=0,~~~~df_{j_1}\wedge\ldots\wedge df_{j_p}=0,$$
for each subset $\{j_1,\ldots,j_p\}$ of the index set $\{1,\ldots,N\}$. Thus
${\rm nnc}(\Delta)$ is analytic.
\medskip\noindent
c) The analyticity of ${\rm nil}(\Delta)$ follows from Cor.~7.7
applied to the plurisubharmonic function $\varphi=\sum 2\lambda_j\,
\log|f_j|$. Assume first that $\lambda_j<1$ and that $\Delta$ has normal 
crossings at $x$. Let $f_{j_1}(x)=\ldots=f_{j_s}(x)=0$ and
\hbox{$f_j(x)\ne 0$} for $j\ne j_l$. Then, we can choose local coordinates 
$(w_1,\ldots,w_n)$ on $U$ such that $w_1=f_{j_1}(z)$, $\ldots$,
$w_s=f_{j_s}(z)$, and we have
$$\int_{U}{d\lambda(z)\over\prod|f_j(z)|^{2\lambda_j}}
\le\int_{U}{C\,d\lambda(w)\over
|w_1|^{2\lambda_1}\ldots|w_s|^{2\lambda_s}}<+\infty.$$
It follows that ${\rm nil}(\Delta)\subset{\rm nnc}(\Delta)$.
Let us prove now the statement relating ${\rm nil}(\Delta)$ with
multiplicity sets. Near any point $x$, we have $|f_j(z)|\le C_j|z-x|^{m_j}$
with $m_j={\rm ord}_x f_j$, thus
$$\prod|f_j|^{-2\lambda_j}\ge C\,|z-x|^{-2\mu(\Delta,x)}.$$
It follows that $x\in{\rm nil}(\Delta)$ as soon as $\mu(\Delta,x)\ge n$.
On the other hand, we are going to prove that $\mu(\Delta,x)<1$ implies
$x\notin{\rm nil}(\Delta)$, i.e.\ $\prod|f_j|^{-2\lambda_j}$ integrable 
near $x$. We may assume $\lambda_j$ rational; otherwise replace each
$\lambda_j$ by a slightly larger rational number in such a way that
$\mu(\Delta,x)<1$ is still true. Set $f=\prod f_j^{k\lambda_j}$
where $k$ is a common denominator. The result is then a consequence of the
following lemma.\qed
\endproof

\claim{(8.2) Lemma} If $f\in\cO_{X,x}$ is not identically $0$, there exists
a neighbor\-hood $U$ of $x$ such that $\int_U|f|^{-2\lambda}\,dV$ converges
for all $\lambda<1/m$, $m={\rm ord}_x f$.
\endclaim

\proof{Proof} One can assume that $f$ is a Weierstrass polynomial
$$f(z)=z_n^m+a_1(z')z_n^{m-1}+\cdots+a_m(z'),~~~~a_j(z')\in\cO_{n-1},~~~~
a_j(0)=0,$$
with respect to some coordinates $(z_1,\ldots,z_n)$ centered at $x$.
Let $v_j(z')$, \hbox{$1\le j\le m$,} denote the roots $z_n$ of $f(z)=0$.
On a small neighborhood $U$ of $x$ we have $|v_j(z')|\le 1$.
The inequality between arithmetic and geometric mean implies
$$\eqalign{
\int_{\{|z_n|\le 1\}}|f(z)|^{-2\lambda}\,dx_ndy_n&=\int_{\{|z_n|\le 1\}}
\prod_{1\le j\le m}|z_n-v_j(z')|^{-2\lambda}\,dx_ndy_n\cr
&\le{1\over m}\int_{\{|z_n|\le 1\}}\sum_{1\le j\le m}|z_n-v_j(z')
|^{-2m\lambda}\,dx_ndy_n\cr
&\le\int_{\{|z_n|\le 2\}}{dx_ndy_n\over|z_n|^{2m\lambda}},\cr}$$
so the Lemma follows from the Fubini theorem.\qed
\endproof

Another interesting application concerns the study of multiplicities
of singular points for algebraic hypersurfaces in $\bP^n$.
Following \cite{Waldschmidt~1975}, we introduce the following definition.

\claim{(8.3) Definition} Let $S$ be a finite subset of $\bP^n$. For any 
integer $t\ge 1$, we define $\omega_t(S)$ as the minimum of the degrees of 
non zero homogeneous polynomials $P\in\bC[z_0,\ldots,z_n]$ which
vanish at order $t$ at every point of $S$, i.e.\ $D^\alpha P(w)=0$
for every $w\in S$ and every multi-index $\alpha=(\alpha_0,\ldots,\alpha_n)$ of 
length $|\alpha|<t$.
\endclaim

It is clear that $t\longmapsto\omega_t(S)$~ is a non-decreasing and
subadditive function, i.e.\ for all integers $t_1,t_2\ge1$
we have $\omega_{t_1+t_2}(S)\le\omega_{t_1}(S)+\omega_{t_2}(S)$. One defines
$$\Omega(S)=\inf_{t\ge 1}{\omega_t(S)\over t}.\leqno(8.4)$$
For all integers $t,t'\ge 1$, the monotonicity and subadditivity of
$\omega_t(S)$ show that 
$$\omega_t(S)\le([t/t']+1)\,\omega_{t'}(S),~~~~{\rm hence}~~
\Omega(S)\le{\omega_t(S)\over t}\le\Big({1\over t'}+{1\over t}\Big)\,
\omega_{t'}(S).$$
We find therefore
$$\Omega(S)=\lim_{t\to+\infty}{\omega_t(S)\over t}.\leqno(8.5)$$
Our goal is to find a lower bound of $\Omega(S)$ in terms of $\omega_t(S)$.
For $n=1$, it is obvious that $\Omega(S)=\omega_t(S)/t={\rm card}\,S$
for all $t$. From now on, we assume that~$n\ge 2$.

\claim{(8.6) Theorem} Let $t_1,t_2\ge 1$ be integers, let $P$ be a homogeneous 
poly\-nomial of degree $\omega_{t_2}(S)$ vanishing at order
$\ge t_2$ at every point of~$S$. If $P=P_1^{k_1}\ldots P_N^{k_N}$ is
the decomposition of $P$ in irreducible factors and
\hbox{$Z_j=P_j^{-1}(0)$}, we set
$$\alpha={t_1+n-1\over t_2},~~~~\Delta=\sum(k_j\alpha-[k_j\alpha])\,[Z_j],
~~~~a=\dim\big({\rm nil}(\Delta)\big).$$
Then we have the inequality
$${\omega_{t_1}(S)+n-a-1\over t_1+n-1}\le{\omega_{t_2}(S)\over t_2}.$$
\endclaim

Let us first make a few comments before giving the proof.
If we let $t_2$ tend to infinity and observe that
${\rm nil}(\Delta)\subset{\rm nnc}(\Delta)$ by Prop.~8.1~c), we get
$a\le 2$ and
$${\omega_{t_1}(S)+1\over t_1+n-1}\le\Omega(S)\le
{\omega_{t_2}(S)\over t_2}.\leqno(8.7)$$
Such a result was first obtained by \cite{Waldschmidt~1975, 1979} with the
lower bound $\omega_{t_1}(S)/(t_1+n-1)$, as a consequence of the
H\"ormander-Bombieri-Skoda theorem. The above improved inequalities
were then found by (Esnault-Viehweg~1983), who used rather
deep tools of algebraic geometry. Our proof will consist in a
refinement of the Bombieri-Waldschmidt method due to \cite{Azhari~1990}.
It has been conjectured by \cite{Chudnovsky~1979} that
$\Omega(S)\ge(\omega_1(S)+n-1)/n$. Chudnovsky's conjecture is  true for
$n=2$ (as shown by (8.7)); this case was first verified independently
by \cite{Chudnovsky~1979} and \cite{Demailly~1982}. The conjecture can also be
verified in case $S$ is a complete polytope, and the lower bound of
the conjecture is then optimal (see \cite{Demailly 1982a} and ??.?.?).
More generally, it is natural to ask whether the inequality
$${\omega_{t_1}(S)+n-1\over t_1+n-1}\le\Omega(S)\le{\omega_{t_2}(S)\over t_2}
\leqno(8.8)$$
always holds; this is the case if there are infinitely many $t_2$ for
which $P$ can be chosen in such a way that ${\rm nil}(\Delta)$ has
dimension $a=0$.

\claim{(8.9) Bertini's lemma} If $E\subset\bP^n$ is an analytic
subset of dimension $a$, there exists a dense subset in the grassmannian 
of $k$-codimensional linear subspaces $Y$ of $\bP^n$ such that
$\dim(E\cap Y)\le a-k$ $($when $k>a$ this means that $E\cap Y=\emptyset\,)$.
\endclaim

\proof{Proof} By induction on $n$, it suffices to show that $\dim(E\cap H)\le
a-1$ for a generic hyperplane $H\subset\bP^n$. Let $E_j$ be the (finite)
family of irreducible components of $E$, and $w_j\in E_j$ an
arbitrary point. Then $E\cap H=\bigcup E_j\cap H$ and we have 
$\dim E_j\cap H<\dim E_j\le a$ as soon as $H$ avoids all points $w_j$.\qed
\endproof

\proof{Proof of Theorem 8.6.} By Bertini's lemma, there exists a linear
subspace $Y\subset\bP^n$ of codimension $a+1$ such that 
${\rm nil}(\Delta)\cap Y=\emptyset$. We consider $P$ as a section of the 
line bundle $\cO(D)$ over $\bP^n$, where $D=\deg\,P$~
(cf.\ Th.~V-15.5). There are sections $\sigma_1,\ldots,\sigma_{a+1}$
of $\cO(1)$ such that $Y=\sigma^{-1}(0)$. We shall apply Th.~7.1
to $E=\cO(1)$ with its standard hermitian metric, and to $L=\cO(k)$ 
equipped with the additional weight $\varphi=\alpha\log|P|^2$.
We may assume that the open set $U=\{|\sigma|<1\}$ is such that
${\rm nil}(\Delta)\cap\ol U=\emptyset$, otherwise it suffices to multiply 
$\sigma$ by a large constant. This implies that the polynomial
$Q=\prod P_j^{[k_j\alpha]}$ satisfies 
$$\int_U|Q|^2\,e^{-\varphi}\,dV=
\int_U\prod|P_j|^{-2(k_j\alpha-[k_j\alpha])}\,dV<+\infty.$$
Set $\omega=ic\big(\cO(1)\big)$. We have $\ii d'd''\log|P|^2\ge-ic\big(\cO(D)
\big)=-D\omega$ by the Lelong-Poincar\'e equation, thus
$\ii\Theta(L_\varphi)\ge(k-\alpha D)\omega$. The desired curvature inequality
$\ii\Theta(L_\varphi)\ge(a+1+\varepsilon)\ii\Theta(E)$ is satisfied if
$k-\alpha D\ge (a+1+\varepsilon)$. We thus take 
$$k=[\alpha D]+a+2.$$
The section $f\in H^0(U,K_{\bP^n}\otimes L)=H^0\big(U,\cO(k-n-1)\big)$
is taken to be a multiple of $Q$ by some polynomial. This is possible
provided that 
$$k-n-1\ge\deg\,Q~~~\Longleftrightarrow~~~
\alpha D+a+2-n-1\ge\sum[k_j\alpha]\,\deg\,P_j,$$
or equivalently, as $D=\sum k_j\,\deg\,P_j$,
$$\sum(k_j\alpha-[k_j\alpha])\,\deg\,P_j\ge n-a-1.\leqno(8.10)$$
Then we get $f\in H^0(U,K_{\bP^n}\otimes L)$ such that
$\int_U |f|^2\,e^{-\varphi}\,dV<+\infty$. Theorem 7.1 implies the
existence of $F\in H^0(\bP^n,K_{\bP^n}\otimes L)$, i.e.\ of
a polynomial $F$ of degree $k-n-1$, such that 
$$\int_{\bP^n}|F|^2e^{-\varphi}\,dV=\int_{\bP^n}{|F|^2\over|P|^{2\alpha}}
\,dV<+\infty\,;$$
observe that $|\sigma|$ is bounded, for we are on a compact manifold.
Near any $w\in S$, we have $|P(z)|\le C|z-w|^{t_2}$, thus
\hbox{$|P(z)|^{2\alpha}\le C|z-w|^{2(t_1+n-1)}$}. This implies that the 
above integral can converge only if $F$ vanishes at order $\ge t_1$
at each point $w\in S$. Therefore
$$\omega_{t_1}(S)\le\deg\,F=k-n-1=
[\alpha D]+a+1-n\le\alpha\omega_{t_2}(S)+a+1-n,$$
which is the desired inequality.

However, the above proof only works under the additional assumption (8.10).
Assume on the contrary that
$$\beta=\sum(k_j\alpha-[k_j\alpha])\,\deg\,P_j<n-a-1.$$
Then the polynomial $Q$ has degree
$$\sum[k_j\alpha]\,\deg\,P_j=\alpha\,\deg\,P-\beta=
\alpha D-\beta,$$
and $Q$ vanishes at every point $w\in S$ with order
$$\eqalign{
{\rm ord}_w Q&\ge\sum[k_j\alpha]\,{\rm ord}_w P_j=
\alpha\sum k_j\,{\rm ord}_w P_j-\sum(k_j\alpha-[k_j\alpha])\,
{\rm ord}_w P_j\cr
&\ge\alpha~{\rm ord}_w P-\beta\ge\alpha t_2-\beta=t_1-(\beta-n+1).\cr}$$
This implies ${\rm ord}_w Q\ge t_1-[\beta-n+1]$. As
$[\beta-n+1]<n-a-1-n+1=-a\le 0$, we can take a derivative
of order $-[\beta-n+1]$ of $Q$ to get a polynomial $F$ with
$$\deg\,F=\alpha D-\beta+[\beta-n+1]\le\alpha D-n+1,$$
which vanishes at order $t_1$ on $S$. In this case, we obtain therefore
$$\omega_{t_1}(S)\le\alpha D-n+1={t_1+n-1\over t_2}\,\omega_{t_2}(S)
-n+1$$
and the proof of Th.~8.6 is complete.\qed
\endproof

\section{9}{Skoda's $L^2$ Estimates for Surjective Bundle Morphisms}
Let $(X,\omega)$ be a K\"ahler manifold, $\dim X=n$, and $g:E\longrightarrow Q$ a
holomorphic morphism of hermitian vector bundles over $X$. Assume
in the first instance that $g$ is {\it surjective}. We are interested in
conditions insuring for example that the induced morphism
$g~:~H^k(X,K_X\otimes E)\longrightarrow H^k(X,K_X\otimes Q)$ is also surjective.
For that purpose, it is natural to consider the subbundle $S=\Ker g\subset E$
and the exact sequence
$$0\longrightarrow S\longrightarrow E\buildo g\over\longrightarrow Q\longrightarrow 0.\leqno(9.1)$$
Assume for the moment that $S$ and $Q$ are endowed with the metrics
induced by that of $E$. Let $L$ be a line bundle over $X$. We consider
the tensor product of sequence (9.1) by $L\,:$
$$0\longrightarrow S\otimes L\longrightarrow E\otimes L\buildo g\over\longrightarrow Q\otimes L\longrightarrow0.\leqno(9.2)$$

\claim{(9.3) Theorem} Let $k$ be an integer such that $0\le k\le n$. Set
$r=\rk\,E$, $q={\rm rk Q}$, $s=\rk\,S=r-q$~ and
$$m=\min\{n-k,s\}=\min\{n-k,r-q\}.$$
Assume that $(X,\omega)$ possesses also a complete K\"ahler metric $\wh\omega$,
that $E\ge_m 0$, and that $L\longrightarrow X$ is a hermitian line bundle such that
$$\ii\Theta(L)-(m+\varepsilon)\ii\Theta(\det Q)\ge 0$$
for some $\varepsilon>0$. Then for every $D''$-closed form $f$ of type 
$(n,k)$ with values in $Q\otimes L$ such that $\|f\|<+\infty$, there exists a 
$D''$-closed form $h$ of type $(n,k)$ with values in $E\otimes L$ such
that $f=g\cdot h$ and 
$$\|h\|^2\le(1+m/\varepsilon)\,\|f\|^2.$$
\endclaim

The idea of the proof is essentially due to \cite{Skoda~1978}, who actually
proved the special case $k=0$. The general case appeared in 
\cite{Demailly~1982c}.

\proof{Proof} Let $j:S\to E$ be the inclusion morphism, $g^\star:Q\to E$
and $j^\star:E\to S$ the adjoints of $g,j$, and
$$D_E = \pmatrix{D_S&-\beta^\star\cr \beta & D_Q\cr},~~
\beta\in \cC^\infty_{1,0}\big(X,\hom(S,Q)\big),~~
\beta^\star\in \cC^\infty_{0,1}\big(X,\hom(Q,S)\big),$$
the matrix of $D_E$ with respect to the orthogonal splitting 
$E\simeq S\oplus Q$ (cf.\ \S V-14). Then $g^\star f$ is a lifting of $f$
in $E\otimes L$. We shall try to find $h$ under the form
$$h=g^\star f+ju,~~~~u\in L^2_{n,k}(X,S\otimes L).$$
As the images of $S$ and $Q$ in $E$ are orthogonal, we have $|h|^2=|f|^2+|u|^2$
at every point of $X$. On the other hand $D''_{Q\otimes L}f=0$ by hypothesis 
and $D''g^\star=-j\circ\beta^\star$ by V-14.3~d), hence
$$D''_{E\otimes L}h=-j(\beta^\star\wedge f)+j\,D''_{S\otimes L}=
j(D''_{S\otimes L}-\beta^\star\wedge f).$$
We are thus led to solve the equation
$$D''_{S\otimes L}u=\beta^\star\wedge f,\leqno(9.4)$$
and for that, we apply Th.~4.5 to the $(n,k+1)$-form 
$\beta^\star\wedge f$. One observes now that the curvature of $S\otimes L$
can be expressed in terms of $\beta$. This remark will be used to prove:
\endproof

\claim{(9.5) Lemma} $\langle A_k^{-1}(\beta^\star\wedge f),
(\beta^\star\wedge f)\rangle\le(m/\varepsilon)\,|f|^2$.
\endclaim

If the Lemma is taken for granted, Th.~4.5 yields a solution $u$ of~(9.4)
in $L^2_{n,q}(X,S\otimes L)$ such that $\|u\|^2\le(m/\varepsilon)\,\|f\|^2$.
As \hbox{$\|h\|^2=\|f\|^2+\|u\|^2$}, the
proof of Th.~9.3 is complete.\qed

\proof{Proof of Lemma~9.5.} Exactly as in the proof of Th.~VII-10.3,
formulas (V-14.6) and (V-14.7) yield
$$\ii\Theta(S)\ge_m \ii\beta^\star\wedge\beta,~~~~
\ii\Theta(\det Q)\ge\Tr_Q(\ii\beta\wedge
\beta^\star)=\Tr_S(-\ii\beta^\star\wedge\beta).$$
Since $\cC^\infty_{1,1}(X,\Herm\,S)\ni\Theta:=-\ii\beta^\star\wedge\beta
\ge_\Grif0$, Prop.~VII-10.1 implies
$$m\,\Tr_S(-\ii\beta^\star\wedge\beta)\otimes\Id_S+\ii\beta^\star\wedge
\beta\ge_m 0.$$
From the hypothesis on the curvature of $L$ we get
$$\eqalign{
\ii\Theta(S\otimes L)&\ge_m \ii\Theta(S)\otimes\Id_L+(m+\varepsilon)\,
\ii\Theta(\det Q)\otimes\Id_{S\otimes L}\cr
&\ge_m\big(\ii\beta^\star\wedge\beta+(m+\varepsilon)\,\Tr_S(-\ii\beta^\star
\wedge\beta)\otimes\Id_S\big)\otimes\Id_L\cr
&\ge_m(\varepsilon/m)\,(-\ii\beta^\star\wedge\beta)\otimes\Id_S\otimes
\Id_L.\cr}$$
For any $v\in\Lambda^{n,k+1}T^\star_X\otimes S\otimes L$, Lemma~VII-7.2 implies
$$\langle A_{k,S\otimes L}v,v\rangle\ge(\varepsilon/m)\,
\langle -\ii\beta^\star\wedge\beta\wedge\Lambda v,v\rangle,\leqno(9.6)$$
because $\rk(S\otimes L)=s$ and $m=\min\{n-k,s\}$. Let $(dz_1,\ldots,dz_n)$
be an orthonormal basis of $T^\star_X$ at a given point $x_0\in X$ and set 
$$\beta=\sum_{1\le j\le n}dz_j\otimes\beta_j,~~~~\beta_j\in\hom(S,Q).$$
The adjoint of the operator~ $\beta^\star\wedge\bu=\sum d\ol z_j\wedge
\beta_j^\star\,\bu$ ~is the contraction $\beta\ort \bu$ defined by
$$\beta\ort v=\sum{\partial\over\partial\ol z_j}\ort (\beta_j v)=
\sum -\ii dz_j\wedge\Lambda(\beta_j v)=-\ii\beta\wedge\Lambda v.$$
We get consequently $\langle -\ii\beta^\star\wedge\beta\wedge\Lambda v,v\rangle
=|\beta\ort v|^2$ and (9.6) implies
$$|\langle\beta^\star\wedge f,v\rangle|^2=|\langle f,\beta\ort v\rangle|^2\le
|f|^2\,|\beta\ort v|^2\le(m/\varepsilon)
\langle A_{k,S\otimes L}v,v\rangle\,|f|^2.\eqno{\square}$$
\endproof

If $X$ has a plurisubharmonic exhaustion function $\psi$, we can select
a convex increasing function $\chi\in \cC^\infty(\bR,\bR)$ and
multiply the metric of $L$ by the weight $\exp(-\chi\circ\psi)$
in order to make the $L^2$ norm of $f$ converge. Theorem 9.3 implies 
therefore:

\claim{(9.7) Corollary} Let $(X,\omega)$ be a weakly pseudoconvex K\"ahler 
manifold, let $g:E\to Q$ be a surjective bundle morphism with
$r=\rk\,E$, $q=\rk\,Q$, let \hbox{$m=\min\{n-k,r-q\}$} and let
$L\to X$ be a hermitian line bundle. Suppose that $E\ge_m 0$ and
$$\ii\Theta(L)-(m+\varepsilon)\,\ii\Theta(\det Q)\ge 0$$
for some $\varepsilon>0$. Then $g$ induces a surjective map
$$H^k(X,K_X\otimes E\otimes L)\longrightarrow H^k(X,K_X\otimes Q\otimes L).$$
\endclaim

The most remarkable feature of this result is that it does not require any 
strict positivity assumption on the curvature (for instance $E$ can be a flat
bundle). A careful examination of the proof shows that it amounts to verify 
that the image of the coboundary morphism
$$-\beta^\star\wedge\bu~:~H^k(X,K_X\otimes Q\otimes L)\longrightarrow 
H^{k+1}(X,K_X\otimes S\otimes L)$$
vanishes; however the cohomology group $H^{k+1}(X,K_X\otimes S\otimes L)$
itself does not vanish in general as it would do under a strict positivity
assumption (cf.\ Th.~VII-9.4).

We want now to get also estimates when $Q$ is endowed with a metric given
a priori, that can be distinct from the quotient metric of $E$ by $g$.
Then the map\break 
\hbox{$g^\star(gg^\star)^{-1}\,:\,Q\to E$} is the lifting of $Q$
orthogonal to $S=\Ker g$. The quotient metric $|\bu|'$ on $Q$ is
therefore defined in terms of the original metric $|\bu|$ by
$$|v|^{\prime 2}=|g^\star(gg^\star)^{-1}v|^2=\langle(gg^\star)^{-1}v,v\rangle
=\det(gg^\star)^{-1}\,\langle\wt{gg^\star}v,v\rangle$$
where $\wt{gg^\star}\in\End(Q)$ denotes the endomorphism of $Q$
whose matrix is the transposed of the comatrix of $gg^\star$. For every
$w\in\det Q$, we find
$$|w|^{\prime2}=\det(gg^\star)^{-1}\,|w|^2.$$
If $Q'$ denotes the bundle $Q$ with the quotient metric, we get
$$\ii\Theta(\det Q')=\ii\Theta(\det Q)+\ii d'd''\log\det(gg^\star).$$
In order that the hypotheses of Th.~9.3 be satisfied, we are led to define
a new metric $|\bu|'$ on $L$ by $|u|^{\prime 2}=|u|^2\,\big(\det(gg^\star)
\big)^{-m-\varepsilon}$. Then
$$\ii\Theta(L')=\ii\Theta(L)+(m+\varepsilon)\,\ii d'd''\log\det(gg^\star)\ge
(m+\varepsilon)\,\ii\Theta(\det Q').$$
Theorem 9.3 applied to $(E,Q',L')$ can now be reformulated:

\claim{(9.8) Theorem} Let $X$ be a complete K\"ahler manifold equipped with
a K\"ahler metric $\omega$ on~$X$, let $E\to Q$ be
a surjective morphism of hermitian vector bundles and let
$L\to X$ be a hermitian line bundle. Set $r=\rk\,E$,
$q=\rk\,Q$ and $m=\min\{n-k,r-q\}$ and suppose $E\ge_m 0$, 
$$\ii\Theta(L)-(m+\varepsilon)\ii\Theta(\det Q)\ge 0$$
for some $\varepsilon>0$. Then for every $D''$-closed form $f$ of type 
$(n,k)$ with values in $Q\otimes L$ such that 
$$I=\int_X\langle\wt{gg^\star}f,f\rangle\,(\det gg^\star)^{-m-1-\varepsilon}\,
dV<+\infty,$$
there exists a $D''$-closed form $h$ of type $(n,k)$ with values in 
$E\otimes L$ such that $f=g\cdot h$ and 
$$\int_X|h|^2\,(\det gg^\star)^{-m-\varepsilon}\,dV
\le(1+m/\varepsilon)\,I.\eqno\square$$
\endclaim

Our next goal is to extend Th.~9.8 in the case when $g~:~E\longrightarrow Q$
is only {\it generically} surjective; this means that the analytic set
$$Y=\{x\in X~;~g_x~:~E_x\longrightarrow Q_x~{\rm is~not~surjective~}\}$$
defined by the equation $\Lambda^q g=0$ is nowhere dense in $X$.
Here $\Lambda^q g$ is a section of the bundle $\hom(\Lambda^qE,\det Q)$.

\claim{(9.9) Theorem} The existence statement and the estimates of Th.~$9.8$
remain true for a generically surjective morphism $g:E\to Q$
provided that $X$ is weakly pseudoconvex.
\endclaim

\proof{Proof} Apply Th.~9.8 to each relatively compact domain
$X_c\ssm Y$ (these domains are complete K\"ahler by Lemma~7.2).
From a sequence of solutions on $X_c\ssm Y$ we can extract a subsequence 
converging weakly on $X\ssm Y$ as c tends to $+\infty$. One gets a 
form $h$ satisfying the estimates, such that $D''h=0$ on $X\ssm Y$ and
$f=g\cdot h$. In order to see that $D''h=0$ on $X$, it suffices 
to apply Lemma~7.3 and to observe that $h$ has $L^2_\loc$
coefficients on $X$ by our estimates.\qed
\endproof

A very special but interesting case is obtained for the trivial bundles
$E=\Omega\times\bC^r$, $Q=\Omega\times\bC$ over a pseudoconvex open set
\hbox{$\Omega\subset\bC^n$}. Then the morphism $g$ is given by a $r$-tuple
$(g_1,\ldots,g_r)$ of holomorphic functions on $\Omega$. Let us take $k=0$
and $L=\Omega\times\bC$ with the metric given by a weight $e^{-\varphi}$.
If we observe that $\wt{gg^\star}=\Id$ when $\rk\,Q=1$, 
Th.~9.8 applied on \hbox{$X=\Omega\ssm g^{-1}(0)$} and Lemmas~7.2,
7.3 give:

\claim{(9.10) Theorem {\rm(\cite{Skoda 1978})}} Let $\Omega$ be a 
complete K\"ahler open subset of $\bC^n$ and $\varphi$ a
plurisubharmonic function on $\Omega$.  Set $m=\min\{n,r-1\}$. Then
for every holomorphic function $f$ on $\Omega$ such that
$$I=\int_{\Omega\ssm Z}|f|^2\,|g|^{-2(m+1+\varepsilon)}e^{-\varphi}\,dV
<+\infty,$$
where $Z=g^{-1}(0)$, there exist holomorphic functions $(h_1,\ldots,h_r)$
on $\Omega$ such that $f=\sum g_jh_j$ and
$$\int_{\Omega\ssm Y}|h|^2\,|g|^{-2(m+\varepsilon)}e^{-\varphi}\,dV
\le(1+m/\varepsilon)I.\eqno{\square}$$
\endclaim

This last theorem can be used in order to obtain a quick solution of the 
Levi problem mentioned in \S I-4. It can be used also to prove
a result of \cite{Diederich-Pflug~1981}, relating the pseudoconvexity 
property and the existence of complete K\"ahler metrics
for domains of $\bC^n$.

\claim{(9.11) Theorem} Let $\Omega\subset\bC^n$ be an open subset. Then:
\smallskip
\item{\rm a)} $\Omega$ is a domain of holomorphy if and only if $\Omega$
is pseudoconvex~$;$
\smallskip
\item{\rm b)} If $\omcirc=\Omega$ and if $\Omega$ has a complete K\"ahler
metric $\wh\omega$, then $\Omega$ is pseudoconvex.\smallskip
\endclaim

Note that statement b) can be false if the assumption $\omcirc=\Omega$ is
omitted: in fact $\bC^n\ssm\{0\}$ is complete K\"ahler by Lemma~7.2, 
but it is not pseudoconvex if $n\ge 2$.

\proof{Proof} b) By Th.~I-4.12, it is enough to verify that $\Omega$
is a domain of holomorphy, i.e.\ that for every connected open subset $U$
such that $U\cap\partial\Omega\ne\emptyset$ and every connected
component W of $U\cap\Omega$ there exists a holomorphic function $h$
on $\Omega$ such that $h_{\restriction W}$ cannot be continued to~$U$.
Since $\omcirc=\Omega$, the set $U\ssm\ol\Omega$ is not empty. We select
$a\in U\ssm\ol\Omega$. Then the integral 
$$\int_\Omega|z-a|^{-2(n+\varepsilon)}\,dV(z)$$
converges. By Th.~9.10 applied to $f(z)=1$, $g_j(z)=z_j-a_j$ and
$\varphi=0$, there exist holomorphic functions $h_j$ on $\Omega$ such that
$\sum(z_j-a_j)\,h_j(z)=1$. This shows that at least one of the functions
$h_j$ cannot be analytically continued at $a\in U$.
\medskip
\noindent a) Assume that $\Omega$ is pseudoconvex. Given any open connected set
$U$ such that $U\cap\partial\Omega\ne\emptyset$, choose
$a\in U\cap\partial\Omega$. By Th.~I-4.14~c) the function
$$\varphi(z)=(n+\varepsilon)(\log(1+|z|^2)-2\log d(z,\complement\Omega)\big)$$
is plurisubharmonic on $\Omega$. Then the integral
$$\int_\Omega|z-a|^{-2(n+\varepsilon)}\,e^{-\varphi(z)}\,dV(z)
\le\int_\Omega(1+|z|^2)^{-n-\varepsilon}\,dV(z)$$ 
converges, and we conclude as for b).\qed
\endproof

\section{10}{Application of Skoda's $L^2$ Estimates to Local Algebra}
We apply here Th.~9.10 to the study of ideals in the ring
$\cO_n=\bC\{z_1,\ldots,z_n\}$ of germs of holomorphic functions on
$(\bC^n,0)$. Let $\cI=(g_1,\ldots,g_r)\ne(0)$ be an ideal of $\cO_n$.

\claim{(10.1) Definition} Let $k\in\bR_+$.  We associate to $\cI$ the
following ideals:
\smallskip
\item{\rm a)} the ideal $\ol\cI^{(k)}$ of germs $u\in\cO_n$ such that 
$|u|\le C|g|^k$ for some constant $C\ge 0$, where 
$|g|^2=|g_1|^2+\cdots+|g_r|^2$.
\smallskip
\item{\rm b)} the ideal $\wh\cI^{(k)}$ of germs
$u\in\cO_n$ such that
$$\int_\Omega|u|^2\,|g|^{-2(k+\varepsilon)}\,dV<+\infty$$
on a small ball $\Omega$ centered at $0$, if $\varepsilon>0$ is small 
enough.\smallskip
\endclaim

\claim{(10.2) Proposition} For all $k,l\in\bR_+$ we have
\smallskip
\item{\rm a)} $\ol\cI^{(k)}\subset\wh\cI^{(k)}\,;$
\smallskip
\item{\rm b)} $\cI^k\subset\ol\cI^{(k)}$ if $k\in\bN\,;$
\smallskip
\item{\rm c)} $\ol\cI^{(k)}.\ol\cI^{(l)}\subset\ol\cI^{(k+l)}\,;$
\smallskip
\item{\rm d)} $\ol\cI^{(k)}.\wh\cI^{(l)}\subset\wh\cI^{(k+l)}.$
\endclaim

All properties are immediate from the definitions except a)
which is a consequence of Lemma~8.2. Before stating the main result,
we need a simple lemma.

\claim{(10.3) Lemma} If $\cI=(g_1,\ldots,g_r)$ and $r>n$, we can find
elements $\wt g_1,\ldots,\wt g_n\in\cI$ such that $C^{-1}|g|\le|\wt g|\le C|g|$
on a neighborhood of $0$. Each $\wt g_j$ can be taken to be a linear
combination
$$\wt g_j=a_j.\,g=\sum_{1\le k\le r}a_{jk}g_k,~~~~a_j\in\bC^r\ssm\{0\}$$
where the coefficients $([a_1],\ldots,[a_n])$ are chosen in the complement
of a proper analytic subset of $(\bP^{r-1})^n$.
\endclaim

It follows from the Lemma that the ideal $\cJ=(\wt g_1,\ldots,\wt g_n)\subset\cI$
satisfies $\smash{\ol\cJ}^{(k)}=\smash{\ol\cI}^{(k)}$ and
$\smash{\wh\cJ}^{(k)}=\smash{\wh\cI}^{(k)}$ for all $k$.

\proof{Proof} Assume that $g\in\cO(\Omega)^r$. Consider the analytic subsets in
$\Omega\times(\bP^{r-1})^n$ defined by
$$\eqalign{
A&=\big\{(z,[w_1],\ldots,[w_n])\,;\,w_j.\,g(z)=0\big\},\cr
A^\star&=\bigcup{\rm irreducible~components~of}~A~{\rm not~contained~in~}
g^{-1}(0)\times(\bP^{r-1})^n.\cr}$$
For $z\notin g^{-1}(0)$ the fiber $A_z=\{([w_1],\ldots,[w_n])\,;\,w_j.\,g(z)=0\}
=A^\star_z$ is a product of $n$ hyperplanes in $\bP^{r-1}$, hence
$A\cap(\Omega\ssm g^{-1}(0))\times(\bP^{r-1})^n$ is a fiber bundle
with base $\Omega\ssm g^{-1}(0)$ and fiber $(\bP^{r-2})^n$. As
$A^\star$ is the closure of this set in $\Omega\times(\bP^{r-1})^n$,
we have
$$\dim A^\star=n+n(r-2)=n(r-1)=\dim(\bP^{r-1})^n.$$
It follows that the zero fiber
$$A^\star_0=A^\star\cap\big(\{0\}\times(\bP^{r-1})^n\big)$$
is a proper subset of $\{0\}\times(\bP^{r-1})^n$.
Choose $(a_1,\ldots,a_n)\in(\bC^r\ssm\{0\})^n$ such that
\hbox{$(0,[a_1],\ldots,[a_n])$} is not in~$A^\star_0$. By an easy
compactness argument
the set\break \hbox{$A^\star\cap\big(\ol
B(0,\varepsilon)\times(\bP^{r-1})^n\big)$} is disjoint from the
neighborhood 
$$B(0,\varepsilon)\times\prod[B(a_j,\varepsilon)]$$
of \hbox{$(0,[a_1],\ldots,[a_n])$} for $\varepsilon$ small enough. For $z\in
B(0,\varepsilon)$ we have \hbox{$|a_j.\,g(z)|\ge\varepsilon|g(z)|$} for
some~$j$, otherwise the inequa\-lity $|a_j.\,g(z)|<\varepsilon|g(z)|$ would
imply the existence of $h_j\in\bC^r$ with $|h_j|<\varepsilon$ and
$a_j.\,g(z)=h_j.\,g(z)$. Since $g(z)\ne 0$, we would have
$$(z,[a_1-h_1],\ldots,[a_n-h_n])\in A^\star\cap\big(B(0,\varepsilon)\times
(\bP^{r-1})^n\big),$$
a contradiction. We obtain therefore
$$\varepsilon|g(z)|\le\max|a_j.\,g(z)|\le(\max|a_j|)\,|g(z)|~~~~{\rm on}~~
B(0,\varepsilon).\eqno{\square}$$
\endproof

\claim{(10.4) Theorem {\rm(\cite{Brian\c con-Skoda~1974})}}
Set $p=\min\{n-1,r-1\}$. Then
\smallskip
\item{\rm a)} $\wh\cI^{(k+1)}=\cI\,\wh\cI^{(k)}=\ol\cI\,\wh\cI^{(k)}$~~
for $k\ge p$.
\smallskip
\item{\rm b)} $\ol\cI^{(k+p)}\subset\wh\cI^{(k+p)}\subset\cI^k$~~
for all $k\in\bN$.
\endclaim

\proof{Proof} a) The inclusions $\cI\,\wh\cI^{(k)}\subset\ol\cI\,\wh\cI^{(k)}
\subset\wh\cI^{(k+1)}$ are obvious thanks to Prop.~10.2,
so we only have to prove that $\wh\cI^{(k+1)}\subset\cI\,\wh\cI^{(k)}$.
Assume first that $r\le n$. Let $f\in\wh\cI^{(k+1)}$ be such that
$$\int_\Omega|f|^2\,|g|^{-2(k+1+\varepsilon)}\,dV<+\infty.$$
For $k\ge p-1$, we can apply Th.~9.10 with $m=r-1$ and
with the weight $\varphi=(k-m)\log|g|^2$. Hence $f$ can be written 
$f=\sum g_jh_j$ with
$$\int_\Omega|h|^2\,|g|^{-2(k+\varepsilon)}\,dV<+\infty,$$
thus $h_j\in\wh\cI^{(k)}$ and $f\in\cI\,\wh\cI^{(k)}$. When $r>n$,
Lemma~10.3 shows that there is an ideal $\cJ\subset\cI$ with $n$
generators such that $\smash{\wh\cJ}^{(k)}=\smash{\wh\cI}^{(k)}$. We find
$$\wh\cI^{(k+1)}=\wh\cJ^{(k+1)}\subset\cJ\,\wh\cJ^{(k)}\subset
\cI\,\wh\cI^{(k)}~~~~{\rm for}~~k\ge n-1.$$
\medskip
\noindent
b) Property a) implies inductively $\wh\cI^{(k+p)}=\cI^k\,\wh\cI^{(p)}$
for all $k\in\bN$. This gives in particular
$\wh\cI^{(k+p)}\subset\cI^k$.\qed
\endproof

\claim{(10.5) Corollary} \smallskip
\item{\rm a)} The ideal $\ol\cI$ is the integral
closure of $\cI$, i.e.\ by definition the set of germs $u\in\cO_n$ which
satisfy an equation
$$u^d+a_1u^{d-1}+\cdots+a_d=0,~~~~a_s\in\cI^s,~~~~1\le s\le d.$$
\smallskip
\item{\rm b)} Similarly, $\ol\cI^{(k)}$ is the set of germs $u\in\cO_n$ which
satisfy an equation
$$u^d+a_1u^{d-1}+\cdots+a_d=0,~~~~a_s\in\cI^{]ks[},~~~~1\le s\le d,$$
where $]t[$ denotes the smallest integer $\ge t$.\smallskip
\endclaim

As the ideal $\ol\cI^{(k)}$ is finitely generated, property b) shows
that there always exists a rational number $l\ge k$ such that 
$\smash{\ol\cI}^{(l)}=\smash{\ol\cI}^{(k)}$.

\proof{Proof} a) If $u\in\cO_n$ satisfies a polynomial equation with coefficients
$a_s\in\cI^s$, then clearly $|a_s|\le C_s\,|g|^s$
and Lemma~II-4.10 implies $|u|\le C\,|g|$.

Conversely, assume that $u\in\ol\cI$. The ring $\cO_n$ is Noetherian, so
the ideal $\smash{\wh\cI}^{(p)}$ has a finite number of generators 
$v_1,\ldots,v_N$. For every $j$ we have $uv_j\in\ol\cI\,\smash{\wh\cI}^{(p)}=
\cI\,\smash{\wh\cI}^{(p)}$, hence there exist
elements $b_{jk}\in\cI$ such that
$$uv_j=\sum_{1\le k\le N}b_{jk}v_k.$$
The matrix $(u\delta_{jk}-b_{jk})$ has the non zero vector $(v_j)$ in
its kernel, thus $u$ satisfies the equation $\det(u\delta_{jk}-b_{jk})=0$,
which is of the required type.
\medskip
\noindent b) Observe that $v_1,\ldots,v_N$ satisfy simultaneously some
integrability condition\break $\int_\Omega|v_j|^{-2(p+\varepsilon)}<+\infty$,
thus $\smash{\wh\cI}^{(p)}=\smash{\wh\cI}^{(p+\eta)}$ for 
$\eta\in[0,\varepsilon[$. Let $u\in\smash{\ol\cI}^{(k)}$.
For every integer $m\in\bN$ we have
$$u^mv_j\in\ol\cI^{(km)}\,\wh\cI^{(p+\eta)}\subset
\wh\cI^{(km+\eta+p)}.$$
If $k\notin\bQ$, we can find $m$ such that $d(km+\varepsilon/2,\bZ)
<\varepsilon/2$,
thus $km+\eta\in\bN$ for some $\eta\in{}]0,\varepsilon[$. If
$k\in\bQ$, we take $m$ such that $km\in\bN$ and $\eta=0$. Then
$$u^mv_j\in\wh\cI^{(N+p)}=\cI^N\,\wh\cI^{(p)}~~~~{\rm with}~~
N=km+\eta\in\bN,$$
and the reasoning made in a) gives
$\det(u^m\delta_{jk}-b_{jk})=0$ for some \hbox{$b_{jk}\in\cI^N$}. This is an
equation of the type described in b), where the coefficients
$a_s$ vanish when $s$ is not a multiple of $m$ and
$a_{ms}\in\cI^{Ns}\subset\cI^{]kms[}$.\qed
\endproof

Let us mention that Brian\c con and Skoda's result 10.4~b) is optimal for
$k=1$. Take for example $\cI=(g_1,\ldots,g_r)$ with $g_j(z)=z_j^r$,
$1\le j\le r$, and $f(z)=z_1\ldots z_r$. Then $|f|\le C|g|$ and 10.4~b)
yields $f^r\in\cI\,;$ however, it is easy to verify that $f^{r-1}\notin\cI$.
The theorem also gives an answer to the following conjecture made 
by J.~Mather.

\claim{(10.6) Corollary} Let $f\in\cO_n$ and $\cI_f=(z_1\partial f/
\partial z_1,\ldots,z_n\partial f/\partial z_n)$. Then
$f\in\ol\cI_f$, and for every integer $k\ge 0$, $f^{k+n-1}\in\cI^k_f$.
\endclaim

The Corollary is also optimal for $k=1\,:$ for example, one can verify that 
the function $f(z)=(z_1\ldots z_n)^3+z_1^{3n-1}+\ldots+z_n^{3n-1}$ is such that 
$f^{n-1}\notin\cI_f$.

\proof{Proof} Set $g_j(z)=z_j\,\partial f/\partial z_j$, $1\le j\le n$.
By 10.4~b), it suffices to show that $|f|\le C|g|$.
For every germ of analytic curve $\bC\ni t\longmapsto\gamma(t)$,
$\gamma\not\equiv 0$, the vanishing order of $f\circ\gamma(t)$ at $t=0$
is the same as that of
$$t\,{d(f\circ\gamma)\over dt}=\sum_{1\le j\le n}t\,\gamma_j'(t)\,
{\partial f\over\partial z_j}\big(\gamma(t)\big).$$
We thus obtain 
$$|f\circ\gamma(t)|\le C_1\,|t|~\Big|{d(f\circ\gamma)\over dt}\Big|
\le C_2\,\sum_{1\le j\le n}|t\,\gamma_j'(t)|~\Big|
{\partial f\over\partial z_j}\big(\gamma(t)\big)\Big|
\le C_3\,|g\circ\gamma(t)|$$
and conclude by the following elementary lemma.\qed
\endproof

\claim{(10.7) Lemma} Let $f,g_1,\ldots,g_r\in\cO_n$ be germs of holomorphic
functions vanishing at $0$. Then we have $|f|\le C|g|$ for some constant $C$
if and only if for every germ of analytic curve $\gamma$ through $0$ there
exists a constant $C_\gamma$ such that
\hbox{$|f\circ\gamma|\le C_\gamma|g\circ\gamma|$}.
\endclaim

\proof{Proof} If the inequality $|f|\le C|g|$ does not hold on any
neighborhood of $0$, the germ of analytic set $(A,0)\subset(\bC^{n+r},0)$ 
defined by
$$g_j(z)=f(z)z_{n+j},~~~~1\le j\le r,$$
contains a sequence of points $\big(z_\nu,g_j(z_\nu)/f(z_\nu)\big)$
converging to $0$ as $\nu$ tends to $+\infty$, with $f(z_\nu)\ne 0$.
Hence $(A,0)$ contains an irreducible component on which $f\not\equiv 0$
and there is a germ of curve \hbox{$\wt\gamma=(\gamma,\gamma_{n+j}):
(\bC,0)\to(\bC^{n+r},0)$} contained in $(A,0)$ such that $f\circ\gamma
\not\equiv 0$. We get \hbox{$g_j\circ\gamma=(f\circ\gamma)\gamma_{n+j}$},
hence \hbox{$|g\circ\gamma(t)|\le C|t|\,|f\circ\gamma(t)|$} and the
inequality \hbox{$|f\circ\gamma|\le C_\gamma|g\circ\gamma|$} does not
hold.\qed
\endproof

\section{11}{Integrability of Almost Complex Structures}
Let $M$ be a $\cC^\infty$ manifold of real dimension $m=2n$. An {\it almost
complex structure} on $M$ is by definition an endomorphism
$J\in\End(TM)$ of class $\cC^\infty$ such that $J^2=-\Id$.
Then $TM$ becomes a complex vector bundle for which the scalar multiplication
by $i$ is given by $J$. The pair $(M,J)$ is said to be an {\it almost
complex manifold}. For such a manifold, the complexified tangent space
$T_\bC M=\bC\otimes_\bR TM$ splits into conjugate complex subspaces
$$T_\bC M=T^{1,0}M\oplus T^{0,1}M,~~~~\dim_\bC T^{1,0}M=\dim_\bC T^{0,1}M=n,
\leqno(11.1)$$
where $T^{1,0}M$, $T^{0,1}M\subset T_\bC M$ are the eigenspaces of
$\Id\otimes J$ corresponding to the eigenvalues $i$ and $-i$. 
The complexified exterior algebra $\bC\otimes_\bR\Lambda^\bu T^\star M=
\Lambda^\bu T_\bC^\star M$ has a corresponding splitting
$$\Lambda^k T^\star_\bC M=\bigoplus_{p+q=k}\Lambda^{p,q} T_\bC^\star M
\leqno(11.2)$$
where we denote by definition
$$\Lambda^{p,q}T^\star_\bC M=\Lambda^p(T^{1,0}M)^\star\otimes_\bC
\Lambda^q(T^{0,1}M)^\star.\leqno(11.3)$$
As for complex manifolds, we let $C^s_{p,q}(M,E)$ be the
space of differential forms of class $C^s$ and bidegree $(p,q)$ on $M$
with values in a complex vector bundle $E$. There is a natural
antisymmetric bilinear map
$$\theta~:~~\cC^\infty(M,T^{1,0}M)\times \cC^\infty(M,T^{1,0}M)\longrightarrow \cC^\infty(M,T^{0,1}M)$$
which associates to a pair $(\xi,\eta)$ of $(1,0)$-vector fields
the $(0,1)$-component of the Lie bracket $[\xi,\eta]$. Since 
$$[\xi,f\eta]=f[\xi,\eta]+(\xi.f)\,\eta,~~~~\forall f\in \cC^\infty(M,\bC)$$
we see that $\theta(\xi,f\eta)=f\,\theta(\xi,\eta)$. It follows that
$\theta$ is in fact a $(2,0)$-form on $M$ with values in $T^{0,1}M$.

If $M$ is a {\it complex analytic manifold} and $J$ its natural almost
complex structure, we have in fact $\theta=0$, because
$[\partial/\partial z_j,\partial/\partial z_k]=0$, $1\le j,k\le n$,
for any holomorphic local coordinate system $(z_1,\ldots,z_n)$.

\claim{(11.4) Definition} The form $\theta\in \cC^\infty_{2,0}(M,T^{0,1}M)$ is
called the torsion form of $J$. The almost complex structure $J$ is
said to be integrable if $\theta=0$.
\endclaim

\claim{(11.5) Example} \rm If $M$ is of real dimension $m=2$, every
almost complex structure is integrable, because $n=1$ and alternate
$(2,0)$-forms must be zero. Assume that $M$ is a smooth oriented
surface. To any Riemannian metric $g$ we can associate the
endomorphism $J\in\End(TM)$ equal to the rotation of $+\pi/2$. 
A change of orientation changes $J$ into the conjugate structure $-J$.
Conversely, if $J$ is given, $TM$ is a complex line bundle, so $M$ is 
oriented, and a Riemannian metric $g$ is associated to $J$ if
and only if $g$ is $J$-hermitian. As a consequence, there is a one-to-one
correspondence between conformal classes of Riemannian metrics on $M$
and almost complex structures corresponding to a given orientation.\qed
\endclaim

If $(M,J)$ is an almost complex manifold and $u\in \cC^\infty_{p,q}(M,\bC)$, we 
let $d'u,~d''u$ be the components of type $(p+1,q)$ and $(p,q+1)$ in the 
exterior derivative $du$. Let $(\xi_1,\ldots,\xi_n)$ be a 
frame of $T^{1,0}M_{\restriction\Omega}$. The torsion form $\theta$ can be 
written
$$\theta=\sum_{1\le j\le n}\alpha_j\otimes\ol\xi_j,~~~~
\alpha_j\in \cC^\infty_{2,0}(\Omega,\bC).$$
Then $\theta$ yields conjugate operators $\theta',\theta''$ on 
$\Lambda^\bu T_\bC^\star M$ such that
$$\theta'u=\sum_{1\le j\le n}\alpha_j\wedge(\ol\xi_j\ort\,u),~~~~
 \theta''u=\sum_{1\le j\le n}\ol\alpha_j\wedge(\xi_j\ort\,u).\leqno(11.6)$$
If $u$ is of bidegree $(p,q)$, then $\theta'u$ and $\theta''u$ are of
bidegree $(p+2,q-1)$ and $(p-1,q+2)$. It is clear that $\theta'$, $\theta''$
are derivations, i.e.\
$$\theta'(u\wedge v)=(\theta'u)\wedge v+(-1)^{\deg\,u}u\wedge(\theta'v)$$
for all smooth forms $u,v$, and similarly for $\theta''$.  

\claim{(11.7) Proposition} We have~ $d=d'+d''-\theta'-\theta''$.
\endclaim

\proof{Proof} Since all operators occuring in the formula are derivations, it is
sufficient to check the formula for forms of degree 0 or~1. If $u$ is
of degree 0, the result is obvious because $\theta'u=\theta''u=0$ and $du$
can only have components of types $(1,0)$ or $(0,1)$. If $u$ is
a $1$-form and $\xi,\eta$ are complex vector fields, we have
$$du(\xi,\eta)=\xi.u(\eta)-\eta.du(\xi)-u([\xi,\eta]).$$
When $u$ is of type $(0,1)$ and $\xi,\eta$ of type 
$(1,0)$, we find
$$(du)^{2,0}(\xi,\eta)=-u\big(\theta(\xi,\eta)\big)~$$
thus $(du)^{2,0}=-\theta'u$, and of course $(du)^{1,1}=d'u$, 
$(du)^{0,2}=d''u$, $\theta''u=0$
by definition. The case of a $(1,0)$-form $u$ follows by conjugation.\qed
\endproof

Proposition 11.7 shows that $J$ is integrable if and only if 
$d=d'+d''$. In this case, we infer immediately
$$d^{\prime 2}=0,~~~~d'd''+d''d'=0,~~~~d^{\prime\prime 2}=0.$$
For an integrable almost complex structure, we thus have the same formalism 
as for a complex analytic structure, and indeed we shall prove:

\claim{(11.8) Newlander-Nirenberg theorem (1957)} Every integrable almost
complex structure $J$ on $M$ is defined by a unique analytic structure.
\endclaim

The proof we shall give follows rather closely that of
\cite{H\"ormander~1966}, which was itself based on previous ideas of
\cite{Kohn~1963, 1964}. A function $f\in C^1(\Omega,\bC)$, $\Omega\subset M$,
is said to be $J$-holomorphic if $d''f=0$. Let $f_1,\ldots,f_p\in
C^1(\Omega,\bC)$ and let $h$ be a function of class $C^1$ on an open
subset of $\bC^p$ containing the range of $f=(f_1,\ldots,f_p)$. An easy
computation gives
$$d''(h\circ f)=\sum_{1\le j\le p}\Big({\partial h\over\partial z_j}
\circ f\Big)d''f_j+\Big({\partial h\over\partial\ol z_j}\circ f\Big)
\ol{d'f_j},\leqno(11.9)$$
in particular $h\circ f$ is $J$-holomorphic as soon as $f_1,\ldots,f_p$
are $J$-holomorphic and $h$ holomorphic in the usual sense.

Constructing a complex analytic structure on $M$ amounts to show the
existence of $J$-holomorphic complex coordinates $(z_1,\ldots,z_n)$ on
a neighborhood $\Omega$ of every point $a\in M$. Formula (11.9)
then shows that all coordinate changes $h:(z_k)\mapsto(w_k)$ are
holomorphic in the usual sense, so that $M$ is furnished with a complex
analytic atlas. The uniqueness of the analytic structure associated to $J$ is
clear, since the holomorphic functions are characterized by the condition
$d''f=0$. In order to show the existence, we need a lemma.

\claim{(11.10) Lemma} For every point $a\in M$ and every integer $s\ge 1$,
there exist $\cC^\infty$ complex coordinates $(z_1,\ldots,z_n)$ centered at $a$
such that
$$d''z_j={\rm O}(|z|^s),~~~~1\le j\le n.$$
\endclaim

\proof{Proof} By induction on $s$. Let $(\xi^\star_1,\ldots,\xi^\star_n)$ be a
basis of $\Lambda^{1,0}T^\star_\bC M$. One can find complex functions
$z_j$ such that $dz_j(a)=\xi^\star_j$, i.e.\
$$d'z_j(a)=\xi^\star_j,~~~~d''z_j(a)=0.$$
Then $(z_1,\ldots,z_n)$ satisfy the conclusions of the Lemma~for $s=1$.
If $(z_1,\ldots,z_n)$ are already constructed for the integer $s$, we have 
a Taylor expansion
$$d''z_j=\sum_{1\le k\le n}P_{jk}(z,\ol z)\,\ol{d'z_k}+{\rm O}(|z|^{s+1})$$
where $P_{jk}(z,w)$ is a homogeneous polynomial in $(z,w)\in\bC^n\times
\bC^n$ of total degree $s$. As $J$ is integrable, we have
$$\eqalign{
0=d^{\prime\prime 2}z_j&=\sum_{1\le k,l\le n}{\partial P_{jk}\over
\partial z_l}\,d''z_l\wedge\ol{d'z_k}+{\partial P_{jk}\over
\partial\ol z_l}\,\ol{d'z_l}\wedge\ol{d'z_k}+{\rm O}(|z|^s)\cr
&=\sum_{1\le k<l\le n}\Big[{\partial P_{jk}\over\partial\ol z_l}-
{\partial P_{jl}\over\partial\ol z_k}\Big]\,
\ol{d'z_l}\wedge\ol{d'z_k}+{\rm O}(|z|^s)\cr}$$
because $\partial P_{jk}/\partial z_l$ is of degree $s-1$ and $d''z_l={\rm O}
(|z|^s)$. Since the polynomial between brackets is of degree $s-1$, we must
have
$${\partial P_{jk}\over\partial\ol z_l}-{\partial P_{jl}\over\partial\ol z_k}
=0,~~~~\forall j,k,l.$$
We define polynomials $Q_j$ of degree $s+1$ 
$$Q_j(z,\ol z)=\int_0^1\sum_{1\le l\le n}\ol z_l\,P_{jl}(z,t\ol z)\,dt.$$
Trivial computations show that
$$\eqalign{
{\partial Q_j\over\partial\ol z_k}&=\int_0^1
\Big(P_{jk}+\sum_{1\le l\le n}\ol z_l\,{\partial P_{jl}\over\partial\ol z_k}
\Big)(z,t\ol z)\,dt\cr
&=\int_0^1{d\over dt}\Big[t\,P_{jk}(z,t\ol z)\Big]\,dt=P_{jk}(z,\ol z),\cr
d''\big(z_j-Q_j(z,\ol z)\big)&=d''z_j-\sum_{1\le k\le n}
{\partial Q_j\over\partial\ol z_k}\,\ol{d'z_k}-
\sum_{1\le k\le n}{\partial Q_j\over\partial z_k}\,d''z_k\cr
&=-\sum_{1\le k\le n}{\partial Q_j\over\partial z_k}\,d''z_k+{\rm O}(|z|^{s+1})
={\rm O}(|z|^{s+1})\cr}$$
because $\partial Q_j/\partial z_k$ is of degree $s$ and $d''z_l=
{\rm O}(|z|)$. The new coordinates
$$\wt z_j=z_j-Q_j(z,\ol z),~~~~1\le j\le n$$
fulfill the Lemma~at step $s+1$.\qed
\endproof

All usual notions defined on complex analytic manifolds can be extended to
integrable almost complex manifolds. For example, a smooth function $\varphi$
is said to be strictly plurisubharmonic if $\ii d'd''\varphi$ is a positive 
definite $(1,1)$-form. Then $\omega=\ii d'd''\varphi$ is a K\"ahler metric on 
$(M,J)$.  

In this context, all $L^2$ estimates proved in the previous paragraphs still 
apply  to an integrable almost complex manifold; remember that the proof of 
the Bochner-Kodaira-Nakano identity used only Taylor developments of order 
$\le 2$, and the coordinates given by Lemma~11.10 work perfectly well
for that purpose. In particular, Th.~6.5 is still valid.

\claim{(11.11) Lemma} Let $(z_1,\ldots,z_n)$ be coordinates centered at a point
$a\in M$ with $d''z_j={\rm O}(|z|^s)$, $s\ge 3$. Then the functions
$$\psi(z)=|z|^2,~~~~\varphi_\varepsilon(z)=|z|^2+\log(|z|^2+\varepsilon^2),
~~~\varepsilon\in{}]0,1]$$
are strictly plurisubharmonic on a small ball $|z|<r_0$.
\endclaim

\proof{Proof} We have
$$\ii d'd''\psi=\ii\sum_{1\le j\le n}d'z_j\wedge\ol{d'z_j}+d'\ol z_j\wedge
d'' z_j+z_j\,d'd''\ol z_j+\ol z_j\,d'd''z_j.$$
The last three terms are ${\rm O}(|z|^s)$ and the first one is positive 
definite at $z=0$, so the result is clear for $\psi$. Moreover
$$\eqalign{
\ii d'd''\varphi_\varepsilon=\ii d'd''\psi&+i{(|z|^2+\varepsilon^2)\sum
d'z_j\wedge\ol{d'z_j}-\sum\ol z_jd'z_j\wedge\ol{\sum\ol z_jd'z_j}\over
(|z|^2+\varepsilon^2)^2}\cr
&+{{\rm O}(|z|^s)\over|z|^2+\varepsilon^2}+
  {{\rm O}(|z|^{s+2})\over(|z|^2+\varepsilon^2)^2}.\cr}$$
We observe that the first two terms are positive definite, whereas the
remainder is ${\rm O}(|z|)$ uniformly in $\varepsilon$.
\endproof

\proof{Proof of theorem 11.8.} With the notations of the previous
lemmas, consider the pseudoconvex open set 
$$\Omega=\{|z|<r\}=\{\psi(z)-r^2<0\},~~~~r<r_0,$$
endowed with the K\"ahler metric $\omega =\ii d'd''\psi$. Let 
$h\in\cD(\Omega)$ be a cut-off function with $0\le h\le 1$ and $h=1$ on a
neighborhood of $z=0$. We apply Th.~6.5 to the $(0,1)$-forms
$$g_j=d''\big(z_jh(z) \big)\in \cC^\infty_{0,1}(\Omega,\bC)$$
for the weight
$$\varphi(z)=A|z|^2+(n+1)\log|z|^2=\lim_{\varepsilon\to 0}A|z|^2+
(n+1)\log(|z|^2+\varepsilon^2).$$
Lemma 11.11 shows that $\varphi$ is plurisubharmonic for $A\ge n+1$,
and for $A$ large enough we obtain
$$\ii d'd''\varphi+\Ric(\omega)\ge\omega~~~~{\rm on}~~\Omega.$$
By Remark~(6.8) we get a function $f_j$ such that $d''f_j=g_j$ and
$$\int_\Omega|f_j|^2e^{-\varphi}dV\le\int_\Omega|g_j|^2e^{-\varphi}dV.$$
As $g_j=d''z_j={\rm O}(|z|^s)$ and $e^{-\varphi}={\rm O}(|z|^{-2n-2})$
near $z=0$, the integral of $g_j$ converges provided that $s\ge 2$.
Then $\int|f_j(z)|^2|z|^{-2n-2}dV$ converges also at $z=0$. Since
the solution $f_j$ is smooth, we must have $f_j(0)=df_j(0)=0$. We set
$$\wt z_j=z_jh(z)-f_j,~~~~1\le j\le n.$$
Then $\wt z_j$ is $J$-holomorphic and $d\wt z_j(0)=dz_j(0)$, so
$(z_1,\ldots,z_n)$ is a $J$-holomorphic coordinate system at $z=0$.\qed
\endproof

In particular, any Riemannian metric on an oriented 2-dimensional real 
manifold defines a unique analytic structure. This fact will be used in order
to obtain a simple proof of the well-known:

\claim{(11.12) Uniformization theorem} Every simply connected
Riemann surface $X$ is biholomorphic either to $\bP^1$, $\bC$ or
the unit disk $\Delta$.
\endclaim

\proof{Proof} We will merely use the fact that $H^1(X,\bR)=0$.  If $X$ is
compact, then $X$ is a complex curve of genus $0$, so $X\simeq\bP^1$ by
Th.~VI-14.16. On the other hand, the elementary Riemann mapping
theorem says that an open set $\Omega\subset\bC$ with
$H^1(\Omega,\bR)=0$ is either equal to $\bC$ or biholomorphic to the
unit disk.  Thus, all we have to show is that a non compact Riemann
surface $X$ with $H^1(X,\bR)=0$ can be embedded in the complex
plane~$\bC$. 

Let $\Omega_\nu$ be an exhausting sequence of relatively compact
connected open sets with smooth boundary in $X$.  We may assume that
$X\ssm \Omega_\nu$ has no relatively compact connected components,
otherwise we ``fill the holes" of $\Omega_\nu$ by taking the union with
all such components.  We let $Y_\nu$ be the double of the manifold with
boundary $(\smash{\ol\Omega_\nu},\partial\Omega_\nu)$, i.e.\ the union of
two copies of $\ol\Omega_\nu$ with opposite orientations and the
boundaries identified.  Then $Y_\nu$ is a compact oriented surface
without boundary.
\endproof

\claim{(11.13) Lemma} We have $H^1(Y_\nu,\bR)=0$.
\endclaim

\proof{Proof} Let us first compute $H^1_c(\Omega_\nu,\bR)$.  Let $u$ be a
closed 1-form with compact support in $\Omega_\nu$.  By Poincar\'e duality
$H^1_c(X,\bR)=0$, so $u=df$ for some function $f\in\cD(X)$.
As $df=0$ on a neighborhood of $X\ssm\Omega_\nu$ and as
all connected components of this set are non compact, $f$ must be equal
to the constant zero near $X\ssm\Omega_\nu$.  Hence $u=df$ is the
zero class in $H^1_c(\Omega_\nu,\bR)$ and we get $H^1_c(\Omega_\nu,\bR)=
H^1(\Omega_\nu,\bR)=0$.  The exact sequence of the pair 
$(\ol\Omega_\nu,\partial\Omega_\nu)$ yields
$$\bR=H^0(\ol\Omega_\nu,\bR)\longrightarrow H^0(\partial\Omega_\nu,\bR)\longrightarrow
H^1(\ol\Omega_\nu,\partial\Omega_\nu\,;\,\bR)\simeq
H^1_c(\Omega_\nu,\bR)=0,$$
thus $H^0(\partial\Omega_\nu,\bR)=\bR$. Finally, the Mayer-Vietoris
sequence applied to small neighborhoods of the two copies of
$\smash{\ol\Omega_\nu}$ in $Y_\nu$ gives an exact sequence
$$H^0(\ol\Omega_\nu,\bR)^{\oplus 2}\longrightarrow H^0(\partial\Omega_\nu,\bR)\longrightarrow
H^1(Y_\nu,\bR)\longrightarrow H^1(\ol\Omega_\nu,\bR)^{\oplus 2}=0$$
where the first map is onto. Hence $H^1(Y_\nu,\bR)=0$.\qed
\endproof

\proof{Proof End of the proof of the uniformization theorem.}
Extend the almost complex structure of $\smash{\ol\Omega_\nu}$ 
in an arbitrary way to $Y_\nu$, e.g.\ by an extension of a Riemannian
metric. Then $Y_\nu$ becomes a compact Riemann surface of genus
$0$, thus $Y_\nu\simeq\bP^1$ and we obtain in particular a
holomorphic embedding $\Phi_\nu:\Omega_\nu\to\bC$.  Fix a point
$a\in\Omega_0$ and a non zero linear form $\xi^\star\in T_aX$.  We can
take the composition of $\Phi_\nu$ with an affine linear map $\bC\to\bC$
so that $\Phi_\nu(a)=0$ and $d\Phi_\nu(a)=\xi^\star$.  By the
well-known properties of injective holomorphic maps, $(\Phi_\nu)$ is
then uniformly bounded on every small disk centered at $a$, thus also
on every compact subset of $X$ by a connectedness argument.  Hence
$(\Phi_\nu)$ has a subsequence converging towards an injective
holomorphic map $\Phi:X\to\bC$.\qed
\endproof


\chapter{Chapter IX}{Finiteness Theorems for 
$q$-Convex Spaces and Stein Spaces}

\section{1}{Topological Preliminaries}
\subsection{1.A}{Krull Topology of $\cO_n$-Modules}
We shall use in an essential way different kind of topological results.
The first of these concern the topology of modules over a local ring
and depend on the Artin-Rees and Krull lemmas.
Let $R$ be a noetherian local ring; ``local" means that $R$ has a unique
maximal ideal $\gm$, or equivalently, that $R$ has an ideal $\gm$ such
that every element $\alpha\in R\ssm\gm$ is invertible.

\claim{(1.1) Nakayama lemma} Let $E$ be a finitely 
generated $R$-module such that $\gm E=E$. Then $E=\{0\}$.
\endclaim

\proof{Proof} By induction on the number of generators of $E\,$:
if $E$ is generated by $x_1,\ldots,x_p$, the hypothesis $E=\gm E$ shows
that $x_p=\alpha_1x_1+\cdots+\alpha_px_p$ with $\alpha_j\in\gm\,$;
as $1-\alpha_p\in R\ssm\gm$ is invertible, we see that $x_p$
can be expressed in terms of $x_1,\ldots,x_{p-1}$
if $p>1$ and that $x_1=0$ if $p=1$.\qed
\endproof

\claim{(1.2) Artin-Rees lemma} Let $F$ be a finitely generated 
$R$-module and let $E$ be a submodule. There exists an integer $s$ such that
$$E\cap\gm^k F=\gm^{k-s}(E\cap\gm^sF)~~~~\hbox{\rm for}~~k\ge s.$$
\endclaim

\proof{Proof} Let $R_t$ be the polynomial ring $R[\gm t]=R+\gm
t+\cdots+\gm^k t^k+\cdots$ where $t$ is an indeterminate. If $g_1,\ldots,
g_p$ is a set of generators of the ideal $\gm$ over $R$, we see that
the ring $R_t$ is generated by $g_1t,\ldots,g_pt$ over $R$, hence $R_t$ is
also noetherian. Now, we consider the $R_t$-modules
$$E_t=\bigoplus E\,t^k,~~~~F_t=\bigoplus~(\gm^k F)\,t^k.$$
Then $F_t$ is generated over $R_t$ by the generators of $F$ over $R$,
hence the submodule $E_t\cap F_t$ is finitely generated. Let $s$ be the 
highest exponent of $t$ in a set of generators $P_1(t),\ldots,P_N(t)$ of
$E_t\cap F_t$. If we identify the components of $t^k$ in the extreme
terms of the equality
$$\bigoplus~\big(E\cap\gm^kF\big)\,t^k=E_t\cap F_t=
\sum_j\Big(\bigoplus_k\gm^k\,t^k\Big)P_j(t),$$
we get
$$E\cap\gm^kF\subset\sum_{l\le s}\gm^{k-l}(E\cap\gm^lF)\subset
\gm^{k-s}(E\cap\gm^sF).$$
The opposite inclusion is clear.\qed
\endproof

\claim{(1.3) Krull lemma} Let $F$ be a finitely generated 
$R$-module and let $E$ be a submodule. Then
\smallskip
\item{\rm a)} $\bigcap_{k\ge 0}\gm^k F=\{0\}$.
\smallskip
\item{\rm b)} $\bigcap_{k\ge 0}(E+\gm^k F)=E$.
\endclaim

\proof{Proof} a) Put $G=\bigcap_{k\ge 0}\gm^k F\subset F$. By the Artin-Rees
lemma, there exists $s\in\bN$ such that $G\cap\gm^k F=
\gm^{k-s}(G\cap\gm^sF)$. Taking $k=s+1$, we find $G\subset\gm G$, hence
$\gm G=G$ and $G=\{0\}$ by the Nakayama lemma.
\medskip
\noindent b) By applying a) to the quotient module $F/E$ we get
$\bigcap\gm^k(F/E)=\{0\}$. Property b) follows.\qed
\endproof

Now assume that $R=\cO_n=\bC\{z_1,\ldots,z_n\}$ and $\gm=(z_1,\ldots,z_n)$.
Then $\cO_n/\gm^k$ is a finite dimensional vector space generated
by the monomials $z^\alpha$, $|\alpha|<k$. It follows that $E/\gm^kE$
is a finite dimensional vector space for any finitely generated
$\cO_n$-module $E$. As $\bigcap\gm^kE=\{0\}$ by 1.3~a),
there is an injection
$$E\lhra\prod_{k\in\bN}E/\gm^kE.\leqno(1.4)$$
We endow $E$ with the Hausdorff topology induced by the product, i.e.\
with the weakest topology that makes all projections $E\longrightarrow E/\gm^k E$
continuous for the complex vector space topology on $E/\gm^k E$. This
topology is called the {\it Krull topology} (or rather, the {\it analytic
Krull topology}; the ``algebraic'' Krull topology would be obtained by
taking the discrete topology on $E/\gm^kE$). For $E=\cO_n$, this is the
topology of simple convergence on coefficients, defined by the
collection of semi-norms $\sum c_\alpha z^\alpha\longmapsto|c_\alpha|$.
Observe that this topology is not complete: the completion of $\cO_n$
can be identified with the ring of formal power series $\bC[[z_1,\ldots,
z_n]]$. In general, the completion is the inverse limit $\wh E=
\smash{\displaystyle \lim_{\longleftarrow}}~E/\gm^k E$.
Every \hbox{$\cO_n$-homomorphism} $E\longrightarrow F$ is continuous, because the 
induced finite dimensional linear maps $E/\gm^k E\longrightarrow F/\gm^k F$ are 
continuous.

\claim{(1.5) Theorem} Let $E\subset F$ be finitely generated $\cO_n$-modules.
Then:
\smallskip
\item{\rm a)} The map $F\longrightarrow G=F/E$ is open, i.e.\ the Krull topology of
$G$ is the quotient of the Krull topology of $F\,;$
\smallskip
\item{\rm b)} $E$ is closed in $F$ and the topology induced by $F$ on
$E$ coincides with the Krull topology of $E$.\smallskip
\endclaim

\proof{Proof} a) is an immediate consequence of the fact that the surjective
finite dimensional linear maps $F/\gm^k F\longrightarrow G/\gm^k G$ are open.
\medskip
\noindent{b)} Let $\ol E$ be the closure of $E$ in $F$. The image of $\ol E$
in $F/\gm^k F$ is mapped into the closure of the image of $E$. As every
subspace of a finite dimensional space is closed, the images of $E$ and
$\ol E$ must coincide, i.e.\ $\ol E+\gm^kF=E+\gm^k F$. Therefore
$$E\subset\ol E\subset\bigcap~(E+\gm^k F)=E$$
thanks to 1.3~b). The topology induced by $F$ on $E$ is the
weakest that makes all projections $E\longrightarrow E/E\cap\gm^k F$ continuous
(via the injections $E/E\cap\gm^kF\lhra F/\gm^kF\,$). However, the 
Artin-Rees lemma gives
$$\gm^kE\subset E\cap\gm^kF=\gm^{k-s}(E\cap\gm^sF)\subset\gm^{k-s}E~~~~
\hbox{\rm for}~~k\ge s,$$
so the topology induced by $F$ coincides with that induced by 
$\prod E/\gm^kE$.\qed
\endproof

\subsection{1.B}{Compact Perturbations of Linear Operators}
We now recall some basic results in the perturbation theory of linear
operators. These results will be needed in order to obtain a finiteness 
criterion for cohomology groups.

\claim{(1.6) Definition} Let $E,F$ be Hausdorff locally convex topological 
vector spaces and $g:E\longrightarrow F$ a continuous linear operator.
\smallskip
\item{\rm a)} $g$ is said to be compact if there exists a neighborhood
$U$ of $0$ in $E$ such that the image $\ol{g(U)}$ is compact in $F$.
\smallskip
\item{\rm b)} $g$ is said to be a monomorphism if $g$ is a topological
isomorphism of $E$ onto a closed subspace of $F$, and a quasi-monomorphism
if $\ker g$ is finite dimensional and $\wt g:E/\ker g\longrightarrow F$ a
monomorphism.
\smallskip
\item{\rm c)} $g$ is said to be an epimorphism if $g$ is surjective 
and open, and a quasi-epimorphism if $g$ is an epimorphism of $E$
onto a closed finite codimensional subspace $F'\subset F$.
\smallskip
\item{\rm d)} $g$ is said to be a quasi-isomorphism if $g$ is
simultaneously a quasi-monomorphism and a quasi-epimorphism.\smallskip
\endclaim

\claim{(1.7) Lemma} Assume that $E,F$ are Fr\'echet spaces. Then
\smallskip
\item{\rm a)} $g$ is a $($quasi-$)$ monomorphism if and only if $g(E)$ is
closed in $F$ and $g$ is injective $($resp.\ and $\ker g$ is 
finite dimensional$\,)$.
\smallskip
\item{\rm b)} $g$ is a $($quasi-$)$ epimorphism if and only if $g$ is 
surjective $($resp.\ $g(E)$ is finite codimen\-sional$\,)$.\smallskip
\endclaim

\proof{Proof} a) If $g(E)$ is closed, the map $\wt g:E/\ker g\longrightarrow g(E)$ is a 
continuous bijective linear map between Fr\'echet spaces, so $\wt g$ is a
topological isomorphism by Banach's theorem.
\medskip
\noindent{b)} If $g$ is surjective, Banach's theorem implies that $g$
is open, thus $g$ is an epimorphism. If $g(E)$ is finite codimensional, let 
$S$ be a supplementary subspace of $g(E)$ in $F$, $\dim S<+\infty$.
Then the map
$$G~:~~(E/\ker g)\oplus S\longrightarrow F,~~~~\wt x\oplus y\longmapsto \wt g(\wt x)+y$$
is a continuous bijective linear map between Fr\'echet spaces, so it is 
a topological isomorphism. In particular $g(E)=G\big((E/\ker g)\oplus\{0\}\big)$
is closed as an image of a closed subspace. Hence $g(E)$
is also a Fr\'echet space and $g:E\longrightarrow g(E)$ is an epimorphism.\qed
\endproof

\claim{(1.8) Theorem {\rm(L.~Schwartz)}} 
Let $h:E\longrightarrow F$ be a compact linear operator.
\smallskip
\item{\rm a)} If $g:E\longrightarrow F$ is a quasi-monomorphism, then $g+h$ is
a quasi-monomorphism.
\smallskip
\item{\rm b)} If $E,F$ are Fr\'echet spaces and if $g:E\longrightarrow F$ is a 
quasi-epimorphism, then $g+h$ is a quasi-epimorphism.\smallskip
\endclaim

\proof{Proof} Set $f=g+h$ and let $U$ be an open convex symmetric neighbor\-hood of 
$0$ in $E$ such that $K=\ol{h(U)}$ is compact.
\medskip
\noindent{a)} It is sufficient to show that there is a closed finite 
codimensional subspace $E'\subset E$ such that $f_{\restriction E'}$ is 
a monomorphism. If we first replace $E$ by a closed supplementary 
subspace of $\ker g$, we see that we may assume $g$ injective. 
Then $g$ is a monomorphism, so we may assume in fact 
that $E$ is a subspace of $F$ and that $g$ is the inclusion. Let $V$ be an
open convex symmetric neighborhood of $0$ in $F$ such that $V\cap E\subset U$.
There exists a closed finite codimensional subspace $F'\subset F$ such that
$K\cap F'\subset 2^{-1}V$ because $\bigcap_{F'}K\cap F'=\{0\}$. If we replace
$E$ by $E'=h^{-1}(F')$ and $U$ by $U'=U\cap E'$, we get
$$K':=\ol{h(U')}\subset K\cap F'\subset 2^{-1}V.$$ 
Hence, we may assume without
loss of generality that $K\subset 2^{-1}V$. Then we show that $f=g+h$
is actually a monomorphism. If $\Omega$ is an arbitrary open
neighborhood of $0$ in $E$, we have to check that there exists a
neighborhood $W$ of $0$ in $F$ such that $f(x)\in W\Longrightarrow x\in
\Omega$. There is an integer $N$ such that $2^{-N}K\cap E\subset\Omega$.
We choose $W$ convex and so small that
$$(W+2^{-N}K)\cap E\subset\Omega~~~{\rm and}~~~2^NW+K\subset 2^{-1}V.$$
Let $x\in E$ be such that $f(x)\in W$. Then $x\in 2^nU$ for some $n\in\bZ$ 
(just take $n\ge n_0$ large enough). For any such $n\ge -N$ we infer
$$x=g(x)=f(x)-h(x)\in W+2^nK=2^n(2^{-n}W+K)\subset 2^{n-1}V.$$
Thus $x\in 2^{n-1}(V\cap E)\subset 2^{n-1}U$. By induction we finally get
$x\in 2^{-N}U$, so
$$x\in(W+2^{-N}K)\cap E\subset\Omega.$$
\noindent{b)} By Lemma~1.7 b), we only have to show that there is a finite
dimensional subspace $S\subset F$ such that the induced map
$$\wt f:E\longrightarrow F\longrightarrow F/S$$
is surjective. If we take $S$ equal to a supplementary subspace of $g(E)$
and replace $g,h$ by the induced maps $\wt g,\wt h:E\longrightarrow F/S$, we may 
assume that $g$ itself is surjective. Then $g$ is open, so $V=g(U)$
is a convex open neighborhood of $0$ in $F$. As $K$ is compact,
there exists a finite set of elements $b_1,\ldots,b_N\in K$ such that 
$K\subset\bigcup(b_j+2^{-1}V)$. If we now take $S={\rm Vect}(b_1,\ldots,b_N)$
and replace again $g,h$ by $\wt g,\,\wt h$,
we obtain $\wt K\subset 2^{-1}\wt V$ where $\wt K$ is the closure of 
$\wt h(U)$ and $V=\wt g(U)$, so we may assume in addition that $K\subset 
2^{-1}V$. Then we show that $f=g+h$ is actually surjective. Let $y_0\in V$.
There exists $x_0\in U$ such that $g(x_0)=y_0$, thus
$$y_1=y_0-f(x_0)=-h(x_0)\in K\subset 2^{-1}V.$$
By induction, we construct $x_n\in 2^{-n}U$ such that $g(x_n)=y_n$ and 
$$y_{n+1}=y_n-f(x_n)=-h(x_n)\in 2^{-n}K\subset 2^{-n-1}V.$$
Hence $y_{n+1}=y_0-f(x_0+\cdots+x_n)$ tends to $0$ in $F$
(as $y_{n+1}\in 2^{-n}K$), but we still
have to make sure that the series $\sum x_n$ converges in $E$.
Let $U_p$ be a fundamental system of convex neighborhoods of $0$ in 
$E$ such that $U_{p+1}\subset 2^{-1}U_p$. For each $p$, $K$ is
contained in the union of the open sets $g(2^nU_p\cap 2^{-1}U)$ when
$n\in\bN$, equal to $g(2^{-1}U)=2^{-1}V$. There exists an increasing
sequence $N(p)$ such that $K\subset g(2^{N(p)}U_p\cap 2^{-1}U)$, thus
$$2^{1-n}K\subset g(2^{N(p)+1-n}U_p\cap 2^{-n}U).$$
As $y_n\in 2^{1-n}K$, we see that we can choose 
$x_n\in2^{N(p)+1-n}\,U_p\cap2^{-n}\,U$ for $N(p)<n\le N(p+1)\,$; then 
$$x_{N(p)+1}+\cdots+x_{N(p+1)}\in(1+2^{-1}+\cdots~~)~U_p\subset 2\,U_p.$$
As $E$ is complete, the series $x=\sum x_n$ converges towards an element 
$x\in E$ such that $f(x)=y_0$, and $f$ is surjective.\qed
\endproof

The following important finiteness theorem due to H.~Cartan and J.-P.\ Serre
can be easily deduced from this.

\claim{(1.9) Theorem} Let $(E^\bu,d)$ and $(F^\bu,\delta)$ be
complexes of Fr\'echet spaces with continuous differentials, and
$\rho^\bu:E^\bu\longrightarrow F^\bu$ a continuous complex morphism. If $\rho^q$
is compact and $H^q(\rho^\bu):H^q(E^\bu)\longrightarrow H^q(F^\bu)$ surjective, then
$H^q(F^\bu)$ is a Hausdorff finite dimensional space.
\endclaim

\proof{Proof} Consider the operators
$$\eqalign{
&g,h~:~~Z^q(E^\bu)\oplus F^{q-1}\longrightarrow Z^q(F^\bu),\cr
&g(x\oplus y)=\rho^q(x)+\delta^{q-1}(y),~~~~h(x\oplus y)=-\rho^q(x).\cr}$$
As $Z^q(E^\bu)\subset E^q$, $Z^q(F^\bu)\subset F^q$ are closed, all our
spaces are Fr\'echet spaces. Moreover the hypotheses imply that $h$ is compact
and $g$ is surjective since $H^q(\rho^\bu)$ is surjective. Hence $g$ is an
epimorphism and $f=g+h=0\oplus\delta^{q-1}$ is a
quasi-epimorphism by 1.8 b). Therefore $B^q(F^\bu)$ is closed and finite
codimensional in $Z^q(F^\bu)$, thus $H^q(F^\bu)$ is Hausdorff and finite
dimensional.\qed
\endproof

\claim{(1.10) Remark} \rm If $\rho^\bu:E^\bu\longrightarrow F^\bu$ is a continuous morphism
of Fr\'echet complexes and if $H^q(\rho^\bu)$ is surjective, then 
$H^q(\rho^\bu)$ is in fact open, because the above map $g$ is open. 
If $H^q(\rho^\bu)$ is bijective, it follows that $H^q(\rho^\bu)$ is 
necessarily a topological isomorphism (however $H^q(E^\bu)$ and $H^q(F^\bu)$
need not be Hausdorff).\qed
\endclaim

\subsection{1.C}{Abstract Mittag-Leffler Theorem}
We will also need the following abstract Mittag-Leffler theorem,
which is a very efficient tool in order to deal with cohomology
groups of inverse limits.

\claim{(1.11) Proposition} Let $(E^\bu_\nu,\delta)_{\nu\in\bN}$ be a
sequence of Fr\'echet complexes together with morphisms
$E^\bu_{\nu+1}\longrightarrow E^\bu_\nu$. We assume that the image of
$E^\bu_{\nu+1}$ in $E^\bu_\nu$ is dense and we let $E^\bu=\displaystyle
\lim_{\displaystyle\longleftarrow}\,E^\bu_\nu$ be the inverse limit complex.
\smallskip
\item{\rm a)} If all maps $H^q(E^\bu_{\nu+1})\longrightarrow H^q(E^\bu_\nu)$,
$\nu\in\bN$, are surjective, then the limit\break 
$H^q(E^\bu)\longrightarrow H^q(E^\bu_0)$ is surjective.
\smallskip
\item{\rm b)} If all maps $H^q(E^\bu_{\nu+1})\longrightarrow H^q(E^\bu_\nu)$,
$\nu\in\bN$, have a dense range, then\break
$H^q(E^\bu)\longrightarrow H^q(E^\bu_0)$ has
a dense range.
\smallskip
\item{\rm c)} If all maps $H^{q-1}(E^\bu_{\nu+1})\longrightarrow H^{q-1}(E^\bu_\nu)$
have a dense range and all maps\break $H^q(E^\bu_{\nu+1})\longrightarrow H^q(E^\bu_\nu)$
are injective, $\nu\in\bN$, then $H^q(E^\bu)\longrightarrow H^q(E^\bu_0)$ is injective.
\smallskip
\item{\rm d)} Let $\varphi^\bu:F^\bu\longrightarrow E^\bu$ be a morphism
of Fr\'echet complexes that has a dense range. If every map 
$H^q(F^\bu)\longrightarrow H^q(E^\bu_\nu)$ has a dense range, then 
$H^q(F^\bu)\longrightarrow H^q(E^\bu)$ has a dense range.\smallskip
\endclaim

\proof{Proof} If $x$ is an element of $E^\bu$ or of $E^\bu_\mu$, $\mu\ge\nu$,
we denote by $x^\nu$ its canonical image in $E^\bu_\nu$. Let
$d_\nu$ be a translation invariant distance that defines the topology
of $E^\bu_\nu$. After replacement of $d_\nu(x,y)$ by
$$d'_\nu(x,y)=\max_{\mu\le\nu}\big\{d_\mu(x^\mu,y^\mu)\big\},~~~~
x,y\in E^\bu_\nu,$$
we may assume that all maps $E^\bu_{\nu+1}\longrightarrow E^\bu_\nu$ are Lipschitz
continuous with coefficient 1. 
\medskip
\noindent{a)} Let $x_0\in Z^q(E^\bu_0)$ represent a given cohomology class 
$\ol x_0\in H^q(E^\bu_0)$.  We construct by induction a convergent
sequence $x_\nu\in Z^q(E^\bu_\nu)$ such that $\ol x_\nu$ is mapped onto
$\ol x_0$.  If $x_\nu$ is already chosen, we can find by assumption
$x_{\nu+1}\in Z^q(E^\bu_{\nu+1})$ such that $\ol x^\nu_{\nu+1}=\ol
x_\nu$, i.e.\ $x_{\nu+1}^\nu=x_\nu +\delta y_\nu$ for some $y_\nu\in
E^{q-1}_\nu$.  If we replace $x_{\nu+1}$ by $x_{\nu+1}-\delta
y_{\nu+1}$ where $y_{\nu+1}\in\smash{E^{q-1}_{\nu+1}}$ yields 
an approximation $y_{\nu+1}^\nu$ of $y_\nu$, we may assume that
$$\max\{d_\nu(y_\nu,0),d_\nu(\delta y_\nu,0)\}\le 2^{-\nu}.$$
Then $(x_\nu)$ converges to a limit $\xi\in Z^q(E^\bu)$ and we have
$\xi^0=x_0+\delta\sum y_\nu^0$. 
\medskip
\noindent{b)} The density assumption for cohomology groups implies that the map
$$Z^q(E^\bu_{\nu+1})\times E^{q-1}_\nu\longrightarrow Z^q(E^\bu_\nu),~~~~(x_{\nu+1},
y_\nu)\longmapsto x_{\nu+1}^\nu+\delta y_\nu$$
has a dense range.  If we approximate $y_\nu$ by elements coming from
$E^{q-1}_{\nu+1}$, we see that the map $Z^q(E^\bu_{\nu+1}) \longrightarrow
Z^q(E^\bu_\nu)$ has also a dense range.  If $x_0\in Z^q(E^\bu_0)$, we
can find inductively a sequence $x_\nu\in Z^q(E^\bu_\nu)$ such that
$d_\nu(x_{\nu+1}^\nu,x_\nu)\le \varepsilon 2^{-\nu-1}$ for all $\nu$,
thus $(x_\nu)$ converges to an element $\xi\in Z^q(E^\bu)$ such that
$d_0(\xi^0,x_0)\le\varepsilon$ and $Z^q(E^\bu)\longrightarrow Z^q(E^\bu_0)$ has
a dense range.  
\medskip
\noindent{c)} Let $x\in Z^q(E^\bu)$ be such that $\ol x^0\in H^q(E^\bu_0)$ is
zero.  By assumption, the image of $\ol x$ in
$H^q(E^\bu_\nu)$ must be also zero, so we can write $x^\nu=dy_\nu$,
$y_\nu\in E^{q-1}_\nu$.  We have $z_\nu=y_{\nu+1}^\nu-y_\nu\in
Z^{q-1}(E^\bu_\nu)$.  Let $z_{\nu+1}\in Z^{q-1}(E^\bu_{\nu+1})$ be such
that $z_{\nu+1}^\nu$ approximates $z_\nu$.  If we replace $y_{\nu+1}$
by $y_{\nu+1}-z_{\nu+1}$, we still have $x^{\nu+1}=dy_{\nu+1}$ and we
may assume in addition that $d_\nu(y_{\nu+1}^\nu,y_\nu)\le 2^{-\nu}$. 
Then $(y_\nu)$ converges towards an element $y\in E^{q-1}$ such that
$x=dy$, thus $\ol x=0$ and $H^q(E^\bu)\longrightarrow H^q(E^\bu_0)$ is
injective.
\medskip
\noindent{d)} For every class $\ol y\in H^q(E^\bu)$, the hypothesis implies 
the existence of \hbox{a sequence} $x_\nu\in Z^q(F^\bu)$ such that 
$\varphi^q(\ol x_\nu)^\nu$ converges to $\ol y^\nu$, that is,
\hbox{$d_\nu(y^\nu,\varphi^q(x_\nu)^\nu+\delta z_\nu)$} tends to $0$ for some
sequence $z_\nu\in E^{q-1}_\nu$. Approximate $z_\nu$ by
$\varphi^{q-1}(w_\nu)^\nu$ for some $w_\nu\in F^{q-1}$ and
replace $x_\nu$ by $x'_\nu=x_\nu+\delta w_\nu$. Then
$\varphi^q(x'_\nu)$ converges to $y$ in $Z^q(E^\bu)$.\qed
\endproof

\section{2}{$q$-Convex Spaces}
\subsection{2.A}{$q$-Convex Functions}
The concept of $q$-convexity, first introduced in \cite{Rothstein~1955},
and further developed by \cite{Andreotti-Grauert~1962}, generalizes the 
concepts of pseudoconvexity already considered in chapters 1 and 8.
Let $M$ be a complex manifold, $\dim_\bC M=n$. A function $v\in C^2(M,\bR)$
is said to be strongly (resp.\ weakly) $q$-convex at a point $x\in M$
if $id'd''v(x)$ has at least $(n-q+1)$ strictly positive (resp.\ nonnegative)
eigenvalues, or equivalently if there exists a $(n-q+1)$-dimensional subspace
$F\subset T_xM$ on which the complex Hessian $H_x v$ is positive definite
(resp.\ semi-positive). Weak 1-convexity is thus equivalent to
plurisubharmonicity. Some authors use different conventions for
the number of positive eigenvalues in $q$-convexity. The reason why
we introduce the number $n-q+1$ instead of $q$ is mainly due to the 
following result:

\claim{(2.1) Proposition} If $v\in C^2(M,\bR)$ is strongly $($weakly$)$
$q$-convex and if $Y$ is a submanifold of $M$, then 
$v_{\restriction Y}$ is strongly $($weakly$)$ $q$-convex.
\endclaim

\proof{Proof} Let $d=\dim Y$. For every $x\in Y$, there exists $F\subset T_xM$
with $\dim F=n-q+1$ such that $Hv$ is (semi-) positive on $F$.
Then $G=F\cap T_xY$ has dimension $\ge (n-q+1)-(n-d)=d-q+1$,
and $H(v_{\restriction Y})$ is (semi-) positive on $G\subset T_xY$. Hence
$v_{\restriction Y}$ is strongly (weakly) $q$-convex at $x$.\qed
\endproof

\claim{(2.2) Proposition} Let $v_j\in C^2(M,\bR)$ be a weakly $($strongly$)$
$q_j$-convex function, $1\le j\le s$, and $\chi\in C^2(\bR^s,\bR)$ 
a convex function that is increasing $($strictly increasing$)$ in all variables.
Then $v=\chi(v_1,\ldots,v_s)$ is weakly $($strongly$)$ $q$-convex with 
$q-1=\sum(q_j-1)$. In particular $v_1+\cdots+v_s$ is weakly
$($strongly$)$ $q$-convex.
\endclaim

\proof{Proof} A simple computation gives
$$Hv=\sum_j{\partial\chi\over\partial t_j}(v_1,\ldots,v_s)\,Hv_j+
\sum_{j,k}{\partial^2\chi\over\partial t_j\partial t_k}(v_1,\ldots,v_s)\,
d'v_j\otimes\ol{d'v_k},\leqno(2.3)$$
and the second sum defines a semi-positive hermitian form.
In every tangent space $T_xM$ there exists a subspace $F_j$ of codimension 
$q_j-1$ on which $Hv_j$ is semi-positive (positive definite). Then 
$F=\bigcap F_j$ has codimension $\le q-1$ and $Hv$ is semi-positive 
(positive definite) on $F$.\qed
\endproof

The above result cannot be improved, as shown by the trivial example
$$v_1(z)=-2|z_1|^2+|z_2|^2+|z_3|^2,~~~~v_2(z)=|z_1|^2-2|z_2|^2+
|z_3|^2~~~\hbox{\rm on}~~\bC^3,$$
in which case $q_1=q_2=2$ but $v_1+v_2$ is only 3-convex.
However, formula (2.3) implies the following result.

\claim{(2.4) Proposition} Let $v_j\in C^2(M,\bR)$, $1\le j\le s$, be such
that every convex linear combination $\sum\alpha_jv_j$, $\alpha_j\ge 0$,
$\sum\alpha_j=1$, is weakly $($strongly$)$ $q$-convex. If 
$\chi\in C^2(\bR^s,\bR)$ is a convex function that is increasing 
$($strictly increasing$)$ in all variables, then $\chi(v_1,\ldots,v_s)$ is weakly 
$($strongly$)$ $q$-convex.\qed
\endclaim

The invariance property of Prop.~2.1 immediately suggests the 
definition of $q$-conve\-xity on complex spaces or analytic schemes:

\claim{(2.5) Definition} Let $(X,\cO_X)$ be an analytic scheme.
A function $v$ on $X$ is said to be strongly $($resp.\ weakly$)$ 
$q$-convex of class $\cC^k$ on $X$ if $X$ can be covered by patches
$G:U\buildo\raise-1.5pt\hbox{$\scriptstyle\simeq$}\over\longrightarrow A$, 
$A\subset\Omega\subset\bC^N$ such that for each
patch there exists a function $\wt v$ on $\Omega$ with $\wt v_{\restriction A}
\circ G=v_{\restriction U}$,
which is strongly $($resp.\ weakly$)$ $q$-convex of class $\cC^k$.
\endclaim

The notion of $q$-convexity on a patch $U$ does not depend on the
way $U$ is embedded in $\bC^N$, as shown by the following lemma.

\claim{(2.6) Lemma} Let $G:U\longrightarrow A\subset\Omega\subset\bC^N$ and
$G':U'\longrightarrow A'\subset\Omega'\subset\bC^{N'}$ be two patches of $X$. 
Let $\wt v$ be a strongly $($weakly$)$ $q$-convex function on $\Omega$
and $v=\wt v_{\restriction A}\circ G$. For every $x\in U\cap U'$ there 
exists a strongly $($weakly$)$ \hbox{$q$-convex}
 function $\wt v'$ on a neighborhood $W'\subset\Omega'$ of $G'(x)$
such that $\wt v'_{\restriction A'\cap W'}\circ G'$ coincides with $v$ on
$G^{\prime-1}(W')$.
\endclaim

\proof{Proof} The isomorphisms
$$\eqalign{
&G'\circ G^{-1}:A\supset G(U\cap U')\longrightarrow G'(U\cap U')\subset A'\cr
&G\circ G^{\prime-1}:A'\supset G'(U\cap U')\longrightarrow G(U\cap U')\subset A\cr}$$
are restrictions of holomorphic maps $H:W\longrightarrow\Omega'$, $H':W'\longrightarrow\Omega$
defined on neighborhoods $W\ni G(x)$, $W'\ni G'(x)\,$; we can shrink
$W'$ so that $H'(W')\subset W$. If we compose the automorphism
$(z,z')\longmapsto(z,z'-H(z))$ of $W\times\bC^{N'}$ with the function
$v(z)+|z'|^2$ we see that the function $\varphi(z,z')=\wt v(z)+|z'-H(z)|^2$ 
is strongly (weakly) $q$-convex on $W\times\Omega'$. Now, $W'$
can be embedded in $W\times\Omega'$ via the map $z'\longmapsto
\big(H'(z'),z'\big)$, so that the composite function
$$\wt v'(z')=\varphi\big(H'(z'),z'\big)=\wt v\big(H'(z')\big)+
|z'-H\circ H'(z')|^2$$
is strongly (weakly) $q$-convex on $W'$ by Prop.~2.1. 
Since $H\circ G=G'$ and $H'\circ G'=G$ on $G^{\prime-1}(W')$, we have
$\wt v'\circ G'=\wt v\circ G=v$ on $G^{\prime-1}(W')$ and the lemma 
follows.\qed
\endproof

A consequence of this lemma is that Prop.~2.2 is still valid for
an analytic scheme $X$ (all the extensions $\wt v_j$ near a given point
$x\in X$ can be obtained with respect to the same local embedding).

\claim{(2.7) Definition} An analytic scheme $(X,\cO_X)$ is said to be 
strongly
$($resp.\ weakly$)$ $q$-convex if $X$ has a $\cC^\infty$ exhaustion function
$\psi$ which is strongly $($resp.\ weakly$)$ $q$-convex outside an
exceptional compact set $K\subset X$. We say that $X$ is strongly 
$q$-complete if $\psi$ can be chosen so that $K=\emptyset$. 
By convention, a compact scheme $X$ is said to be strongly
$0$-complete, with exceptional compact set $K=X$.
\endclaim

We consider the sublevel sets
$$X_c=\{x\in X~;~\psi(x)<c\},~~~~c\in\bR.\leqno(2.8)$$
If $K\subset X_c$, we may select a convex increasing function $\chi$
such that $\chi=0$ on $]-\infty,c]$ and $\chi'>0$ on $]c,+\infty[$.
Then $\chi\circ\psi=0$ on $X_c$, so that $\chi\circ\psi$ is weakly
$q$-convex everywhere in virtue of (2.3). In the weakly
$q$-convex case, we may therefore always assume $K=\emptyset$. 
The following properties are almost immediate consequences of the 
definition:

\claim{(2.9) Theorem} \smallskip
\item{\rm a)} A scheme $X$ is strongly $($weakly$)$ 
$q$-convex if and only if the reduced space $X_\red$ is strongly 
$($weakly$)$ $q$-convex.
\smallskip
\item{\rm b)} If $X$ is strongly $($weakly$)$ $q$-convex, every closed
analytic subset $Y$ of $X_\red$ is strongly $($weakly$)$ $q$-convex.
\smallskip
\item{\rm c)} If $X$ is strongly $($weakly$)$ $q$-convex, every sublevel
set $X_c$ containing the exceptional compact set $K$ is strongly 
$($weakly$)$ $q$-convex.
\smallskip
\item{\rm d)} If $U_j$ is a weakly $q_j$-convex open subset of $X$,
$1\le j\le s$, the intersection $U=U_1\cap\ldots\cap U_s$ is
weakly $q$-convex with $q-1=\sum(q_j-1)$~$;$ $U$ is strongly
$q$-convex $($resp.\ $q$-complete$)$ as soon as one of the sets $U_j$ is 
strongly $q_j$-convex $($resp.\ $q_j$-complete$)$.\smallskip
\endclaim

\proof{Proof} a) is clear, since Def.~2.5 does not involve the structure
sheaf $\cO_X$. In cases b) and c), let $\psi$ be an exhaustion of the
required type on $X$. Then $\psi_{\restriction Y}$ and $1/(c-\psi)$ are
exhaustions on $Y$ and $X_c$ respectively (this is so only if $Y$ is 
closed). Moreover, these functions are strongly (weakly) $q$-convex
on $Y\ssm(K\cap Y)$ and $X_c\ssm K$, thanks to 
Prop.~2.1 and 2.2. For property d), note that a sum
$\psi=\psi_1+\cdots+\psi_s$ of exhaustion functions on the sets $U_j$ is an
exhaustion on $U$, choose the $\psi_j$'s weakly $q_j$-convex 
everywhere, and apply Prop.~2.2.\qed
\endproof

\claim{(2.10) Corollary} Any finite intersection $U=U_1\cap\ldots\cap U_s$
of weakly $1$-convex open subsets is weakly $1$-convex. 
The set $U$ is strongly $1$-convex $($resp.\ $1$-complete$)$ as soon
as one of the sets $U_j$ is strongly $1$-convex $($resp.\ 
$1$-complete$)$.
\endclaim

\subsection{2.B}{Neighborhoods of $q$-complete subspaces}
We prove now a rather useful result asserting the existence of
$q$-complete neighborhoods for $q$-complete subvarieties. The case 
$q=1$ goes back to \cite{Siu~1976}, who used a much more complicated method.
The first step is an approximation-extension theorem for strongly
$q$-convex functions.

\claim{(2.11) Proposition} Let $Y$ be an analytic set in a complex space
$X$ and $\psi$ a strongly $q$-convex $\cC^\infty$ function on $Y$. For every
continuous function $\delta>0$ on $Y$, there exists a strongly $q$-convex
$\cC^\infty$ function $\varphi$ on a neighborhood $V$ of $Y$ such that
$\psi\le\varphi_{\restriction Y}<\psi+\delta$.
\endclaim

\proof{Proof} Let $Z_k$ be a stratication of $Y$ as given by Prop.~II.5.6,
i.e.\ $Z_k$ is an increasing sequence of analytic subsets of $Y$ such that
$Y=\bigcup Z_k$ and $Z_k\ssm Z_{k-1}$ is a smooth $k$-dimensional
manifold (possibly empty for some $k$'s). We shall prove by induction on
$k$ the following statement:

{\it There exists a $\cC^\infty$ function $\varphi_k$ on $X$ which is strongly
$q$-convex along $Y$ and on a closed neighborhood $\ol V_k$ of $Z_k$
in $X$, such that $\psi\le\varphi_{k\restriction Y}<\psi+\delta$.}

We first observe that any smooth extension $\varphi_{-1}$ of $\psi$ to
$X$ satisfies the requirements with $Z_{-1}=V_{-1}=\emptyset$.  Assume that
$V_{k-1}$ and $\varphi_{k-1}$ have been constructed.  Then $Z_k\ssm
V_{k-1}\subset Z_k\ssm Z_{k-1}$ is contained in $Z_{k,\reg}$. 
The closed set $Z_k\ssm V_{k-1}$ has a locally finite covering 
$(A_\lambda)$ in $X$ by open coordinate patches
$A_\lambda\subset\Omega_\lambda\subset\bC^{N_\lambda}$ in which
$Z_k$ is given by equations $z'_\lambda=(z_{\lambda,k+1},\ldots,
z_{\lambda,N_\lambda})=0$. Let $\theta_\lambda$ be $\cC^\infty$ functions
with compact support in $A_\lambda$ such that $0\le\theta_\lambda
\le 1$ and $\sum\theta_\lambda=1$ on $Z_k\ssm V_{k-1}$.  We set
$$\varphi_k(x)=\varphi_{k-1}(x)+\sum~\theta_\lambda(x)\,\varepsilon_\lambda^3
\,\log(1+\varepsilon_\lambda^{-4}|z'_\lambda|^2)~~~~\hbox{\rm on}~~X.$$ 
For $\varepsilon_\lambda>0$ small enough, we will have
$\psi\le\varphi_{k-1\restriction Y}\le\varphi_{k\restriction Y}<
\psi+\delta$. Now, we check that $\varphi_k$ is still strongly
$q$-convex along $Y$ and near any $x_0\in\ol V_{k-1}$,
and that $\varphi_k$ becomes strongly $q$-convex near any
$x_0\in Z_k\ssm V_{k-1}$. We may assume that $x_0\in\Supp
\theta_\mu$ for some $\mu$, otherwise $\varphi_k$ coincides with
$\varphi_{k-1}$ in a neighborhood of $x_0$. Select $\mu$ and a 
small neighborhood $W\compact\Omega_\mu$ of $x_0$ such that
\medskip
\item{a)} if $x_0\in Z_k\ssm V_{k-1}$, then
$\theta_\mu(x_0)>0$ and $A_\mu\cap W\compact\{\theta_\mu>0\}\,$;
\smallskip
\item{b)} if $x_0\in A_\lambda$ for some $\lambda$ (there is only a finite
set $I$ of such $\lambda$'s), then
$A_\mu\cap W\compact A_\lambda$ and $z_{\lambda\restriction A_\mu\cap W}$
has a holomorphic extension $\wt z_\lambda$ to $\ol W\,$;
\smallskip
\item{c)} if $x_0\in\ol V_{k-1}$, then $\varphi_{k-1
\restriction A_\mu\cap W}$ has a strongly $q$-convex extension 
$\wt\varphi_{k-1}$ to $\ol W\,;$
\smallskip
\item{d)} if $x_0\in Y\ssm\ol V_{k-1}$, then $\varphi_{k-1
\restriction Y\cap W}$ has a strongly $q$-convex extension 
$\wt\varphi_{k-1}$ to $\ol W\,.$
\medskip
\noindent{}Otherwise take an arbitrary smooth extension 
$\wt\varphi_{k-1}$ of $\varphi_{k-1\restriction A_\mu\cap W}$ to $\ol W$
and let $\smash{\wt\theta_\lambda}$ be an extension of $\theta_{\lambda
\restriction A_\mu\cap W}$ to $\ol W$. Then
$$\wt\varphi_k=\wt\varphi_{k-1}+\sum\wt\theta_\lambda\,\varepsilon_\lambda^3\,
\log(1+\varepsilon_\lambda^{-4}|{\wt z}^{\,\prime}_\lambda|^2)$$
is an extension of $\varphi_{k\restriction A_\mu\cap W}$ to $\ol W$,
resp.\ of $\varphi_{k\restriction Y\cap W}$ to $\ol W$ in case d).
As the function $\log(1+\varepsilon_\lambda^{-4}
|{\wt z}^{\,\prime}_\lambda|^2)$ is plurisubharmonic and as its first
derivative $\langle{\wt z}^{\,\prime}_\lambda,
d{\wt z}^{\,\prime}_\lambda\rangle\,(\varepsilon_\lambda^4+
|{\wt z}^{\,\prime}_\lambda|^2)^{-1}$ is bounded by
$O(\varepsilon_\lambda^{-2})$, we see that
$$id'd''\wt\varphi_k\ge id'd''\wt\varphi_{k-1}-
O({\scriptstyle\sum}\varepsilon_\lambda).$$ 
Therefore, for $\varepsilon_\lambda$ small enough, $\wt\varphi_k$
remains $q$-convex on $\ol W$ in cases c) and d). Since
all functions ${\wt z}^{\,\prime}_\lambda$ vanish along $Z_k\cap W$,
we have
$$id'd''\wt\varphi_k\ge id'd''\wt\varphi_{k-1}
+\sum_{\lambda\in I}\theta_\lambda\,\varepsilon_\lambda^{-1}\,id'd''
|{\wt z}^{\,\prime}_\lambda|^2\ge id'd''\wt\varphi_{k-1}
+\theta_\mu\,\varepsilon_\mu^{-1}\,id'd''|z'_\mu|^2$$
at every point of $Z_k\cap W$.
Moreover $id'd''\wt\varphi_{k-1}$ has at most $(q-1)$-negative
eigenvalues on $TZ_k$ since $Z_k\subset Y$, whereas
$id'd''|z'_\mu|^2$ is positive definite in the normal directions to
$Z_k$ in $\Omega_\mu$.  In case a), we thus find that $\wt\varphi_k$ 
is strongly $q$-convex on $\smash{\ol W}$ for $\varepsilon_\mu$ small 
enough; we also observe that only finitely many conditions are
required on each $\varepsilon_\lambda$ if we choose a locally finite
covering of $\bigcup\Supp\theta_\lambda$ by neighborhoods
$W$ as above. Therefore, for $\varepsilon_\lambda$ small enough,
$\varphi_k$ is strongly $q$-convex on a
neighborhood $\smash{\ol V'_k}$ of $Z_k\ssm V_{k-1}$.  The
function $\varphi_k$ and the set $V_k=V'_k\cup V_{k-1}$ satisfy the
requirements at order $k$.  It is clear that we can choose the sequence
$\varphi_k$ stationary on every compact subset of $X\,$; the limit
$\varphi$ and the open set $V=\bigcup V_k$ fulfill the proposition.\qed
\endproof

The second step is the existence of almost plurisubharmonic functions
having poles along a prescribed analytic set.  By an almost
plurisubharmonic function on a manifold, we mean a function that is
locally equal to the sum of a plurisubharmonic function and of a smooth
function, or equivalently, a function whose complex Hessian has bounded
negative part.  On a complex space, we require that our function can be
locally extended as an almost plurisubharmonic function in the ambient
space of an embedding. 

\claim{(2.12) Lemma} Let $Y$ be an analytic subvariety in a complex 
space $X$. There is an almost plurisubharmonic function $v$ on $X$
such that $v=-\infty$ on $Y$ with logarithmic poles and 
$v\in \cC^\infty(X\ssm Y)$.
\endclaim

\proof{Proof} Since $\cI_Y\subset\cO_X$ is a coherent subsheaf, there is a
locally finite covering of $X$ by patches $A_\lambda$ isomorphic to
analytic sets in balls $B(0,r_\lambda)\subset\bC^{N_\lambda}$, such
that $\cI_Y$ admits a system of generators $g_\lambda=(g_{\lambda,j})$
on a neighborhood of each set $\ol A_\lambda$.  We set
$$\eqalign{
&v_\lambda(z)=\log|g_\lambda(z)|^2-{1\over r^2_\lambda
-|z-z_\lambda|^2}~~~~\hbox{\rm on}~~A_\lambda,\cr
&v(z)=M_{(1,\ldots,1)}\big(\ldots\,,v_\lambda(z),\,\ldots\big)~~~~\hbox{\rm
for}~~\lambda~~\hbox{\rm such that}~~A_\lambda\ni z,\cr}$$ 
where $M_\eta$ is the regularized max function defined in I-3.37.
As the generators $(g_{\lambda,j})$ can be expressed in terms of
one another on a neighborhood of $\ol A_\lambda\cap \ol A_\mu$, we see
that the quotient $|g_\lambda|/|g_\mu|$ remains bounded on this set. 
Therefore none of the values $v_\lambda(z)$ for $A_\lambda\ni z$ and
$z$ near $\partial A_\lambda$ contributes to the value of $v$,
since $1/(r_\lambda^2-|z-z_\lambda|^2)$ tends to $+\infty$ on
$\partial A_\lambda$. It follows that $v$ is
smooth on $X\ssm Y\,$; as each $v_\lambda$ is almost 
plurisubharmonic on $A_\lambda$, we also see that $v$ is almost 
plurisubharmonic on $X$.\qed
\endproof

\claim{(2.13) Theorem} Let $X$ be a complex space and $Y$ a strongly 
$q$-complete analytic subset. Then $Y$ has a fundamental family of 
strongly $q$-complete neighborhoods $V$ in $X$.
\endclaim

\proof{Proof} By Prop.~2.11 applied to a strongly
$q$-convex exhaustion of $Y$ and $\delta=1$, there exists a strongly
$q$-convex function $\varphi$ on a neighborhood $W_0$ of $Y$ such that
$\varphi_{\restriction Y}$ is an exhaustion.  Let $W_1$ be a
neighborhood of $Y$ such that $\ol W_1\subset W_0$ and such that
$\varphi_{\restriction\ol W_1}$ is an exhaustion.  We are going to show
that every neighborhood $W\subset W_1$ of $Y$ contains a strongly
$q$-complete neighborhood $V$.  If $v$ is the function given by
Lemma~2.12, we set
$$\wt v=v+\chi\circ\varphi~~~~\hbox{\rm on}~~\ol W$$
where $\chi:\bR\to\bR$ is a smooth convex increasing function.  If
$\chi$ grows fast enough, we get $\wt v>0$ on $\partial W$ and the
$(q-1)$-codimensional subspace on which $id'd''\varphi$ is positive 
definite (in some ambient space) is also positive definite for
$id'd''\wt v$ provided that $\chi'$ be large enough to compensate the
bounded negative part of $id'd''v$. Then $\wt v$ is
strongly $q$-convex.  Let $\theta$ be a smooth convex increasing
function on $]-\infty,0[$ such that $\theta(t)=0$ for $t<-3$ and
$\theta(t)=-1/t$ on $]-1,0[$.  The open set $V=\{z\in W\,;\,\wt v(z)<0\}$
is a neighborhood of $Y$ and $\smash{\wt\psi}=\varphi+\theta\circ\wt v$ 
is a strongly $q$-convex exhaustion of~$V$.\qed
\endproof

\subsection{2.C}{Runge Open Subsets}
In order to extend the classical Runge theorem into an approximation
result for sheaf cohomology groups, we need the concept of
a $q$-Runge open subset.

\claim{(2.14) Definition} An open subset $U$ of a complex space $X$
is said to be $q$-Runge $($resp.\ $q$-Runge complete$)$ in $X$ if 
for every compact subset $L\subset U$
there exists a smooth exhaustion function $\psi$ on $X$ and a sublevel set
$X_b$ of $\psi$ such that $L\subset X_b\compact U$ and
$\psi$ is strongly $q$-convex on $X\ssm\ol X_b$
$($resp.\ on the whole space $X)$.
\endclaim

\claim{(2.15) Example} \rm If $X$ is strongly $q$-complete and if $\psi$
is a strongly $q$-convex exhaustion function of $X$, then every 
sublevel set $X_c$ of $\psi$ is $q$-Runge complete
in $X\,$: every compact set $L\subset X_c$ satisfies
$L\subset X_b\compact X_c$ for some $b<c$. More generally, if
$X$ is strongly $q$-convex and if $\psi$ is strongly $q$-convex
on $X\ssm K$, every sublevel set $X_c$ containing $K$
is $q$-Runge in $X$.
\endclaim

Later on, we shall need the following technical result.

\claim{(2.16) Proposition} Let $Y$ be an analytic subset of a complex
space $X$. If $U$ is a $q$-Runge complete open subset of $Y$
and $L$ a compact subset, there exist a neighborhood $V$ of $Y$ in
$X$ and a strongly $q$-convex exhaustion $\wt\psi$ on $V$
such that $U=Y\cap V$ and $L\subset Y\cap V_b\compact U$ for some 
sublevel set $V_b$ of $\wt\psi$.
\endclaim

\proof{Proof} Let $\psi$ be a strongly $q$-convex exhaustion on $Y$ with
$L\subset\{\psi<b\}\compact U$ as in Def.~2.14.  Then
$L\subset\{\psi<b-\delta\}$ for some number $\delta>0$ and Lemma~2.11
gives a strongly $q$-convex function $\varphi$ on a neighborhood $W_0$
of $Y$ so that $\psi\le\varphi_{\restriction Y}<\psi+\delta$.  The
neighborhood $V$ and the function $\smash{\wt\psi}=
\varphi+\theta\circ\wt v$ constructed in the proof of Th.~2.13 are
the desired ones: we have $\psi\le\smash{\wt\psi_{\restriction Y}}
=\varphi_{\restriction Y}<\psi+\delta$, thus 
$$L\subset Y\cap V_{b-\delta}\subset\{\psi<b\}\compact U.\eqno{\square}$$
\endproof

\section{3}{$q$-Convexity Properties in Top Degrees}
It is obvious by definition that a $n$-dimensional complex manifold $M$ is
strongly $q$-complete for $q\ge n+1$ (an arbitrary smooth function is
then strongly $q$-convex~!). If $M$ is connected
and non compact, \cite{Greene and Wu~1975} have shown that $M$ is strongly 
$n$-complete, i.e.\ there is a smooth exhaustion function $\psi$ on $M$
such that $id'd''\psi$ has at least one positive eigenvalue everywhere.
We need the following lemmas.

\claim{(3.1) Lemma} Let $\psi$ be a strongly $q$-convex function 
on $M$ and $\varepsilon>0$ a given number.
There exists a hermitian metric $\omega$ on $M$
such that the eigenvalues $\gamma_1\le\ldots\le\gamma_n$ of the Hessian
form $id'd''\psi$ with respect to $\omega$ satisfy $\gamma_1\ge-\varepsilon$ 
and $\gamma_q=\ldots=\gamma_n=1$.
\endclaim

\proof{Proof} Let $\omega_0$ be a fixed hermitian metric, $A_0\in \cC^\infty(\End TM)$
the hermitian endomorphism associated to the hermitian form $id'd''\psi$ with
respect to $\omega_0$, and $\gamma^0_1\le\ldots\le\gamma^0_n$ the 
eigenvalues of $A_0$ (or $id'd''\psi$). 
We can choose a function $\eta\in \cC^\infty(M,\bR)$ such that
$0<\eta(x)\le\gamma^0_q(x)$ at each point $x\in M$. Select a positive 
function $\theta\in \cC^\infty(\bR,\bR)$ such that
$$\theta(t)\ge |t|/\varepsilon~~\hbox{\rm for}~~t\le 0,~~~
\theta(t)\ge t~~\hbox{\rm for}~~t\ge 0,~~~\theta(t)=t~~\hbox{\rm for}~~t\ge 1.$$
We let $\omega$ be the hermitian metric defined by the hermitian
endomorphism
$$A(x)=\eta(x)\,\theta[(\eta(x))^{-1}A_0(x)]$$
where $\theta[\eta^{-1}A_0]\in \cC^\infty(\End TM)$ is defined as in 
Lemma~VII-6.2. By construction, the eigenvalues of $A(x)$ are 
$\alpha_j(x)=\eta(x)\theta\big(\gamma^0_j(x)/\eta(x)\big)>0$
and we have
$$\cmalign{
&\alpha_j(x)\ge|\gamma^0_j(x)|/\varepsilon~~~~
&\hbox{\rm for}~~\gamma^0_j(x)\le 0,\cr
&\alpha_j(x)\ge\gamma^0_j(x)~~~~&\hbox{\rm for}~~\gamma^0_j(x)\ge 0,\cr
&\alpha_j(x)=\gamma^0_j(x)~~~~&\hbox{\rm for}~~j\ge q~~~
\big(\hbox{\rm then}~\gamma^0_j(x)\ge\eta(x)\big).\cr}$$
The eigenvalues of $id'd''\psi$ with respect to $\omega$ are
$\gamma_j(x)=\gamma^0_j(x)/\alpha_j(x)$ and they have the required
properties.\qed
\endproof

On a hermitian manifold $(M,\omega)$, we consider the Laplace operator 
$\Delta_\omega$ defined by
$$\Delta_\omega v=\hbox{\rm Trace}_\omega(id'd''v)=\sum_{1\le j,k\le n}
\omega^{jk}(z){\partial^2v\over\partial z_j\partial\ol z_k}\leqno(3.2)$$
where $(\omega^{jk})$ is the conjugate of the inverse matrix of 
$(\omega_{jk})$. Note that $\Delta_\omega$ may differ from the usual
Laplace-Beltrami operator if $\omega$ is not K\"ahler. We say that $v$
is strongly $\omega$-subharmonic if $\Delta_\omega v>0$.  This property
implies clearly that $v$ is strongly $n$-convex; however, as
$$\eqalign{
\Delta_\omega\chi(v_1,\ldots,v_s)=\sum_j&{\partial\chi\over\partial t_j}(v_1,\ldots,
v_s)\,\Delta_\omega v_j\cr
&{}+\sum_{j,k}{\partial^2\chi\over\partial t_j\partial
t_k}(v_1,\ldots,v_s)\,\langle d'v_j,d'v_k\rangle_\omega,\cr}$$
subharmonicity has the advantage of being preserved by all convex increasing 
transformations. Conversely, if $\psi$ is strongly $n$-convex and $\omega$
chosen as in Lemma~3.1 with $\varepsilon$ small enough, we get
$\Delta_\omega\psi\ge 1-(n-1)\varepsilon>0$, thus $\psi$ is strongly
subharmonic for a suitable metric $\omega$. 

\claim{(3.3) Lemma} Let $U,W\subset M$ be open sets such that for every
connected component $U_s$ of $U$ there is a connected component
$W_{t(s)}$ of $W$ such that $W_{t(s)}\cap U_s\ne\emptyset$ and
$W_{t(s)}\ssm\smash{\ol U_s}\ne\emptyset$.  Then there exists a 
function $v\in \cC^\infty(M,\bR)$, $v\ge 0$, with support contained in 
$\smash{\ol U\cup\ol W}$, such that $v$ is strongly
$\omega$-subharmonic and $>0$ on $U$.
\endclaim

\proof{Proof} We first prove that the result is true when $U,W$ are small cylinders
with the same radius and axis. Let $a_0\in M$ be a given
point and $z_1,\ldots,z_n$ holomorphic coordinates centered at $a_0$. 
We set $\Re z_j=x_{2j-1}$, $\Im z_j=x_{2j}$,
$x'=(x_2,\ldots,x_{2n})$ and $\omega=\sum\wt\omega_{jk}(x)dx_j\otimes dx_k$.
Let $U$ be the cylinder $|x_1|<r$,
$|x'|<r$, and $W$ the cylinder $r-\varepsilon<x_1<r+\varepsilon$,
$|x'|<r$. There are constants $c,C>0$ such that
$$\sum\wt\omega^{jk}(x)\xi_j\xi_k\ge c|\xi|^2~~~\hbox{\rm and}~~~
\sum|\wt\omega^{jk}(x)|\le C~~~\hbox{\rm on}~~\ol U.$$
Let $\chi\in \cC^\infty(\bR,\bR)$ be a nonnegative
function equal to $0$ on $]-\infty,-r]\cup[r+\varepsilon,+\infty[$ and 
strictly convex on $]-r,r]$. We take explicitly $\chi(x_1)=(x_1+r)
\exp(-1/(x_1+r)^2\big)$ on $]-r,r]$ and
$$v(x)=\chi(x_1)\exp\big(1/(|x'|^2-r^2)\big)~~~\hbox{\rm on}~~U\cup W,~~~
v=0~~~\hbox{\rm on}~~M\ssm(U\cup W).$$
We have $v\in \cC^\infty(M,\bR)$, $v>0$ on $U$, and a simple computation gives
$$\eqalign{
&{\Delta_\omega v(x)\over v(x)}=
\wt\omega^{11}(x)\big(4(x_1+r)^{-5}-2(x_1+r)^{-3}\big)\cr
&~~+\sum_{j>1}\wt
\omega^{1j}(x)\big(1+2(x_1+r)^{-2}\big)(-2x_j)(r^2-|x'|^2)^{-2}\cr 
&~~+\sum_{j,k>1}\wt\omega^{jk}(x)\Big( x_jx_k\big(4-8(r^2-|x'|^2)
\big)-2(r^2-|x'|^2)^2\delta_{jk}\Big)(r^2-|x'|^2)^{-4}.\cr}$$
For $r$ small, we get
$$\eqalign{
{\Delta_\omega v(x)\over v(x)}\ge 2c(x_1+r)^{-5}
&-C_1(x_1+r)^{-2}|x'|(r^2-|x'|^2)^{-2}\cr
&+(2c|x'|^2-C_2r^4)(r^2-|x'|^2)^{-4}\cr}$$
with constants $C_1,C_2$ independent of $r$.  
The negative term is bounded by $C_3(x_1+r)^{-4}+c|x'|^2(
r^2-|x'|^2)^{-4}$, hence
$$\Delta_\omega v/v(x)\ge c(x_1+r)^{-5}+(c|x'|^2-C_2r^4)(r^2-|x'|^2)^{-4}.$$
The last term is negative only when $|x'|<C_4r^2$, in which case
it is bounded by $C_5r^{-4}<c(x_1+r)^{-5}$. Hence
$v$ is strongly $\omega$-subharmonic on $U$.

Next, assume that $U$ and $W$ are connected. Then $U\cup W$ is connected. 
Fix a point $a\in W\ssm\ol U$. If $z_0\in U$ is given, we choose a 
path $\Gamma\subset U\cup W$ from $z_0$ to $a$
which is piecewise linear with respect to holomorphic coordinate
patches. Then we can find a finite sequence of
cylinders $(U_j,W_j)$ of the type described above, $1\le j\le N$,
whose axes are segments contained in $\Gamma$, such that 
$$U_j\cup W_j\subset U\cup W,~~~\ol W_j\subset U_{j+1}~~~\hbox{\rm and}~~~
z_0\in U_0,~~~a\in W_N\subset W\ssm\ol U.$$
For each such pair, we have a function $v_j\in \cC^\infty(M)$ with support in
$\ol U_j\cup\ol W_j$, $v_j\ge 0$, strongly $\omega$-subharmonic and $>0$ on
$U_j$. By induction, we can find constants $C_j>0$ such that
$v_0+C_1v_1+\cdots+C_j v_j$ is strongly $\omega$-subharmonic on
$U_0\cup\ldots\cup U_j$ and $\omega$-subharmonic on $M\ssm\ol W_j$. 
Then
$$w_{z_0}=v_0+C_1v_1+\ldots+C_Nv_N\ge 0$$
is $\omega$-subharmonic on $U$ and strongly $\omega$-subharmonic
$>0$ on a neighborhood $\Omega_0$ of the given point $z_0$. Select
a denumerable covering of $U$ by such neighborhoods $\Omega_p$ and set
$v(z)=\sum\varepsilon_pw_{z_p}(z)$ where $\varepsilon_p$ is a sequence
converging sufficiently fast to $0$ so that $v\in \cC^\infty(M,\bR)$. Then $v$
has the required properties.

In the general case, we find for each pair $(U_s,W_{t(s)})$ a function
$v_s$ with support in $\smash{\ol U_s\cup\ol W_{t(s)}}$, strongly
$\omega$-subharmonic and $>0$ on $U_s$.  Any convergent series
$v=\sum\varepsilon_sv_s$ yields a function with the desired
properties.\qed
\endproof

\claim{(3.4) Lemma} Let $X$ be a connected, locally connected and
locally compact topological space. If $U$ is a relatively compact
open subset of $X$, we let $\wt U$ be the union of $U$ with all compact
connected components of $X\ssm U$. Then $\wt U$ is open and
relatively compact in $X$, and $X\ssm\smash{\wt U}$ has only finitely
many connected components, all non compact.
\endclaim

\proof{Proof} A rather easy exercise of general topology. Intuitively, $\wt U$
is obtained by ``filling the holes" of $U$ in $X$.\qed
\endproof

\claim{(3.5) Theorem {\rm(\cite{Greene-Wu~1975})}} Every $n$-dimensional
connected non compact complex manifold $M$ has a strongly subharmonic 
exhaustion function with respect to any hermitian metric $\omega$. 
In particular, $M$ is strongly $n$-complete.
\endclaim

\proof{Proof} Let $\varphi\in \cC^\infty(M,\bR)$ be an arbitrary exhaustion function. 
There exists a sequence of connected smoothly bounded open sets
$\Omega'_\nu\compact M$ such that $\smash{\ol\Omega'_\nu}\subset\Omega'_{\nu+1}$
and $M=\bigcup\Omega'_\nu$.  Let $\Omega_\nu=\smash{\wt\Omega'_\nu}$ be
the relatively compact open set given by Lemma~3.4.  Then
$\smash{\ol\Omega_\nu}\subset\Omega_{\nu+1}$, $M=\bigcup\Omega_\nu$ 
and $M\ssm\Omega_\nu$ has no compact connected component.  We set
$$U_1=\Omega_2,~~~~U_\nu=\Omega_{\nu+1}\ssm\ol{\Omega_{\nu-2}}
~~~\hbox{\rm for}~~\nu\ge 2.$$ 
Then $\partial U_\nu=\partial\Omega_{\nu+1}\cup\partial\Omega_{\nu-2}\,$;
any connected component $U_{\nu,s}$ of $U_\nu$ has its boundary
$\partial U_{\nu,s}\not\subset\partial\Omega_{\nu-2}$, otherwise $\ol
U_{\nu,s}$ would be open and closed in $M\ssm\Omega_{\nu-2}$,
hence $\ol U_{\nu,s}$ would be a compact component of
$M\ssm\Omega_{\nu-2}$.  Therefore $\partial U_{\nu,s}$ intersects
$\partial\Omega_{\nu+1}\subset U_{\nu+1}$. If $\partial U_{\nu+1,t(s)}$
is a connected component of $U_{\nu+1}$ containing a point of
$\partial U_{\nu,s}$, then $U_{\nu+1,t(s)}\cap U_{\nu,s}\ne
\emptyset$ and $U_{\nu+1,t(s)}\ssm\ol U_{\nu,s}\ne\emptyset$.  
Lemma 7 implies that there is a nonnegative function
$v_\nu\in \cC^\infty(M,\bR)$ with support in $U_\nu\cup U_{\nu+1}$, which is
strongly $\omega$-subharmonic on $U_\nu$.  An induction yields
constants $C_\nu$ such that
$$\psi_\nu=\varphi+C_1v_1+\cdots+C_\nu v_\nu$$ 
is strongly $\omega$-subharmonic on $\ol{\Omega_\nu}\subset
U_0\cup \ldots\cup U_\nu$, thus $\psi=\varphi+\sum C_\nu v_\nu$ is a
strongly $\omega$-subharmonic exhaustion function on
$M$.\qed
\endproof

By an induction on the dimension, the above result can be generalized 
to an arbi\-trary complex space (or analytic scheme), as was first 
shown by T.~Ohsawa.

\claim{(3.6) Theorem {\rm(\cite{Ohsawa~1984})}} Let $X$ be a complex 
space of maximal dimension $n$. 
\smallskip
\item{\rm a)} $X$ is always strongly $(n+1)$-complete.
\smallskip
\item{\rm b)} If $X$ has no compact irreducible component of 
dimension $n$, then $X$ is strongly $n$-complete.
\smallskip
\item{\rm c)} If $X$ has only finitely many irreducible 
components of dimension $n$, then $X$ is strongly $n$-convex.\smallskip
\endclaim

\proof{Proof} We prove a) and b) by induction on $n=\dim X$. For $n=0$, 
property b) is void and a) is obvious (any function can then be
considered as strongly $1$-convex). Assume that a) has been proved
in dimension $\le n-1$. Let $X'$ be the union
of $X_\sing$ and of the irreducible components of $X$ of dimension
at most $n-1$, and $M=X\ssm X'$ the $n$-dimensional part of
$X_\reg$. As $\dim X'\le n-1$, the induction hypothesis shows
that $X'$ is strongly \hbox{$n$-complete}. By Th.~2.13,
there exists a strongly $n$-convex exhaustion function $\varphi'$
on a neighborhood $V'$ of $X'$. Take a closed neighborhood
$\ol V\subset V'$ and an arbitrary exhaustion $\varphi$ on $X$ that extends 
$\varphi'_{\smash{\restriction\ol V}}$. Since every function on a
$n$-dimensional manifold is strongly $(n+1)$-convex, we conclude
that $X$ is at worst $(n+1)$-complete, as stated in a).
\medskip
In case b), the hypothesis means that the connected components $M_j$ 
of $M=X\ssm X'$ have non compact closure $\ol M_j$ in $X$.
On the other hand, Lemma~3.1 shows that there exists a hermitian metric
$\omega$ on $M$ such that $\varphi_{\restriction M\cap V}$ is
strongly $\omega$-subharmonic. Consider the open sets 
$U_{j,\nu}\subset M_j$ provided by Lemma~3.7 below. By the arguments 
already used in Th.~3.5, we can find a strongly $\omega$-subharmonic
exhaustion $\psi=\varphi+\sum_{j,\nu}C_{j,\nu}v_{j,\nu}$ on $X$,
with $v_{j,\nu}$ strongly $\omega$-subharmonic on
$U_{j,\nu}$, $\Supp v_{j,\nu}\subset U_{j,\nu}\cup U_{j,\nu+1}$ and
$C_{j,\nu}$ large. Then $\psi$ is strongly $n$-convex on $X$.
\endproof

\claim{(3.7) Lemma} For each $j$, there exists a sequence of open sets 
$U_{j,\nu}\compact M_j$, $\nu\in\bN$, such that
\smallskip
\item{\rm a)} $M_j\ssm V'\subset\bigcup_\nu U_{j,\nu}$~ and
$(U_{j,\nu})$ is locally finite in $\ol M_j\,;$
\smallskip
\item{\rm b)} for every connected component $U_{j,\nu,s}$ of $U_{j,\nu}$
there is a connected component\break $U_{j,\nu+1,t(s)}$ of $U_{j,\nu+1}$
such that $U_{j,\nu+1,t(s)}\cap U_{j,\nu,s}\ne\emptyset$ and
$U_{j,\nu+1,t(s)}\ssm\smash{\ol U_{j,\nu,s}}\ne\emptyset$.\smallskip
\endclaim

\proof{Proof} By Lemma~3.4 applied to the space $\ol M_j$, there exists a sequence
of relatively compact connected open sets $\Omega_{j,\nu}$ in $\ol M_j$ such 
that $\ol M_j\ssm\Omega_{j,\nu}$ has no compact connected component, 
$\ol\Omega_{j,\nu}\subset\Omega_{j,\nu+1}$ and
$\ol M_j=\bigcup\Omega_{j,\nu}$. We define a compact set 
$K_{j,\nu}\subset M_j$ and an open set $W_{j,\nu}\subset\ol M_j$
containing $K_{j,\nu}$ by
$$K_{j,\nu}=(\ol\Omega_{j,\nu}\ssm\Omega_{j,\nu-1})\ssm V',
~~~~W_{j,\nu}=\Omega_{j,\nu+1}\ssm\ol\Omega_{j,\nu-2}.$$
By induction on $\nu$, we construct an open set
$U_{j,\nu}\compact W_{j,\nu}\ssm X'\subset M_j$ and a finite set 
$F_{j,\nu}\subset \partial U_{j,\nu}\ssm\ol\Omega_{j,\nu}$.
We let $F_{j,-1}=\emptyset$. 
If these sets are already constructed for $\nu-1$, the
compact set $K_{j,\nu}\cup F_{j,\nu-1}$ is contained in the open
set $W_{j,\nu}$, thus contained in a finite union of connected components 
$W_{j,\nu,s}$. We can write $K_{j,\nu}\cup F_{j,\nu-1}=
\bigcup L_{j,\nu,s}$ where $L_{j,\nu,s}$ is contained in 
$W_{j,\nu,s}\ssm X'\subset M_j$. The open set $W_{j,\nu,s}\ssm X'$
is connected and non contained in $\ol\Omega_{j,\nu}\cup
L_{j,\nu,s}$, otherwise its closure $\ol W_{j,\nu,s}$ would have no
boundary point $\in\partial\Omega_{j,\nu+1}$, thus would be open and 
compact in $\ol M_j\ssm\Omega_{j,\nu-2}$, contradiction. 
We select a point $a_s\in (W_{j,\nu,s}\ssm X')\ssm(
\ol\Omega_{j,\nu}\cup L_{j,\nu,s})$ and a smoothly bounded connected 
open set $U_{j,\nu,s}\compact W_{j,\nu,s}\ssm X'$ containing
$L_{j,\nu,s}$ with $a_s\in\partial U_{j,\nu,s}$. 
Finally, we set $U_{j,\nu}=\bigcup_s U_{j,\nu,s}$
and let $F_{j,\nu}$ be the set of all points $a_s$. By construction, we
have $U_{j,\nu}\supset K_{j,\nu}\cup F_{j,\nu-1}$, thus 
$\bigcup U_{j,\nu}\supset\bigcup K_{j,\nu}=M_j\ssm V'$, and $\partial
U_{j,\nu,s}\ni a_s$ with $a_s\in F_{j,\nu}\subset U_{j,\nu+1}$.
Property b) follows.\qed
\endproof

\proof{Proof of Theorem 3.6~c) {\rm (end)}} Let $Y\subset X$ be the
union of $X_\sing$ with all irreducible components of $X$ that are
non compact or of dimension $<n$.  Then $\dim Y\le n-1$, so $Y$ is
$n$-convex and Th.~2.13 implies that there is an exhaustion
function $\psi\in \cC^\infty(X,\bR)$ such that $\psi$ is strongly $n$-convex on
a neighborhood $V$ of $Y$.  Then the complement $K=X\ssm V$ is
compact and $\psi$ is strongly $n$-convex on $X\ssm K$.\qed
\endproof

\claim{(3.8) Proposition} Let $M$ be a connected non compact
$n$-dimensional complex manifold and $U$ an open subset of $M$.
Then $U$ is $n$-Runge complete in $M$ if and only if $M\ssm U$ has no
compact connected component.\qed
\endclaim

\proof{Proof} First observe that a strongly $n$-convex function cannot have
any local maximum, so it satisfies the maximum principle.  If
$M\ssm U$ has a compact connected component $T$, then $T$ has a
compact neighborhood $L$ in $M$ such that $\partial L\subset U$.  We
have $\max_L\psi=\max_{\partial L}\psi$ for every strongly $n$-convex
function, thus $\partial L\subset M_b$ implies $L\subset M_b\,$; thus we
cannot find a sublevel set $M_b$ such that $\partial L\subset M_b\compact U$,
and $U$ is not $n$-Runge in $M$. 

On the other hand, assume that $M\ssm U$ has no compact connected
component and let $L$ be a compact subset of $U$. Let $\omega$ be
any hermitian metric on $M$ and $\varphi$ a strongly $\omega$-subharmonic 
exhaustion function on $M$. Set $b=1+\sup_L\varphi$ and 
$$P=\{x\in M\ssm U\,;\,\varphi(x)\le b\}.$$
As $M\ssm U$ has no compact connected component, all its components
$T_\alpha$ contain a point $y_\alpha$ in 
$$W=\{x\in X\,;\,\varphi(x)>b+1\}.$$
For every point $x\in P$ with $x\in T_\alpha$, there exists a connected open 
set $V_x\compact M\ssm L$ containing $x$ such that $\partial V_x\ni
y_\alpha$ ($M\ssm L$ is a neighborhood of $M\ssm U$ and we can
consider a tubular neighborhood of a path from $x$ to $y_\alpha$ in
$M\ssm L$). The
compact set $P$ can be covered by a finite number of open sets $V_{x_j}$.
Then Lemma~3.3 yields functions $v_j$ with support in 
$\smash{\ol V_{x_j}\cup\ol W}$ which are strongly 
$\omega$-subharmonic on $V_{x_j}$.  Let $\chi$ be a convex increasing
function such that $\chi(t)=0$ on $]-\infty,b]$ and $\chi'(t)>0$ on
$]b,+\infty[$.  Consider the function
$$\psi=\varphi+\sum C_jv_j+\chi\circ\varphi.$$
First, choose $C_j$ large enough so that $\psi\ge b$ on $P$. Then choose
$\chi$ increasing fast enough so that $\psi$ is strongly $\omega$-subharmonic
on $\ol W$. Then $\psi$ is a strongly $n$-convex exhaustion function 
on $M$, and as $\psi\ge\varphi$ on $M$ and $\psi=\varphi$ on $L$, we see that
$$L\subset\{x\in M\,;\,\psi(x)<b\}\subset U.$$
This proves that $U$ is $n$-Runge complete in $M$.\qed
\endproof

\section{4}{Andreotti-Grauert Finiteness Theorems}
\subsection{4.A}{Case of Vector Bundles over Manifolds}
The crucial point in the proof of the Andreotti-Grauert theorems 
is the following special case, which is easily obtained by the
methods of chapter 8.

\claim{(4.1) Proposition} Let $M$ be a strongly $q$-complete 
manifold with $q\ge 1$, and $E$ a holomorphic vector bundle over 
$M$. Then:
\smallskip
\item{\rm a)} $H^k\big(M,\cO(E)\big)=0$~~for $k\ge q$.
\smallskip
\item{\rm b)} Let $U$ be a $q$-Runge complete open subset of $M$.
Every $d''$-closed form $h\in$\break
$\cC^\infty_{0,q-1}(U,E)$ can be approximated uniformly with
all derivatives on every compact subset of $U$ by a sequence of
global $d''$-closed forms $\smash{\wt h_\nu}\in \cC^\infty_{0,q-1}(M,E)$.\smallskip
\endclaim

\proof{Proof} We replace $E$ by 
$\wt E=\Lambda^nTM\otimes E\,$; then we can work with forms of bidegree
$(n,k)$ instead of $(0,k)$. Let $\psi$ be a strongly $q$-convex
exhaustion function on $M$ and $\omega$ the metric given by
Lemma~3.1. Select a function $\rho\in \cC^\infty(M,\bR)$ which increases rapidly 
at infinity so that the hermitian metric $\wt\omega=e^\rho\omega$ is
complete on $M$. Denote by $E_\chi$ the bundle $E$ endowed with the
hermitian metric obtained by multiplication of a fixed metric of $E$ 
by the weight $\exp(-\rho\circ\psi)$ where $\chi\in \cC^\infty(\bR,\bR)$ is a convex
increasing function. We apply Th.~VIII-4.5 for the bundle $E_\chi$ over the
complete hermitian manifold $(M,\wt\omega)$. Then
$$ic(E_\chi)=ic(E)+id'd''(\chi\circ\psi)\otimes\Id_E
\ge_{\rm Nak}ic(E)+\chi'\circ\psi~id'd''\psi\otimes\Id_E.$$
The eigenvalues of $id'd''\psi$ with respect to $\wt\omega$ are
$e^{-\rho}\gamma_j$, so Lemma~VII-7.2 and Prop.~VI-8.3 yield
$$\eqalign{
[ic(E_\chi),\Lambda]+T_{\wt\omega}&\ge[ic(E),\Lambda]+T_{\wt\omega}+
\chi'\circ\psi~[id'd''\psi,\Lambda]\otimes\Id_E\cr
&\ge[ic(E),\Lambda]+T_{\wt\omega}+\chi'\circ\psi~e^{-\rho}(\gamma_1+\cdots+
\gamma_k)\otimes\Id_E\cr}$$
when this curvature tensor acts on $(n,k)$-forms. For $k\ge q$, we have
$$\gamma_1+\cdots+\gamma_k\ge 1-(q-1)\varepsilon>0~~~~\hbox{\rm if}~~
\varepsilon\le 1/q.$$
We choose $\chi_0$ increasing fast enough so that all the eigenvalues 
of the above curvature tensor are $\ge 1$ when $\chi=\chi_0$. 
Then for every $g\in \cC^\infty_{n,k}(M,E)$ with $D''g=0$
the equation $D''f=g$ can be solved with an estimate
$$\int_M|f|^2e^{-\chi\circ\psi}dV\le\int_M|g|^2e^{-\chi\circ\psi}dV,
$$
where $\chi=\chi_0+\chi_1$ and where $\chi_1$ is a convex increasing function
chosen so that the integral of $g$ converges. This gives a).
In order to prove b), let $h\in \cC^\infty_{n,q-1}(U,E)$ be such that $D''h=0$
and let $L$ be an arbitrary compact subset of $U$.
Thanks to Def.~2.14, we can choose $\psi$ such that there is
a sublevel set $M_b$ with $L\subset M_b\compact U$.
Select $b_0<b$ so that $L\subset M_{b_0}$, and let
$\theta\in \cC^\infty(\bR,\bR)$ be a convex increasing function such that
$\theta=0$ on $]-\infty,b_0[$ and $\theta\ge 1$ on $]b,+\infty[$.
Let $\eta\in\cD(U)$ be a cut-off function such that $\eta=1$
on $M_b$. We solve the equation $D''f=g$ for $g=D''(\eta h)$
with the weight $\chi=\chi_0+\nu \theta\circ\psi$ and let $\nu$ tend to
infinity. As $g$ has compact support in $U\ssm M_b$ and
$\chi\circ\psi\ge\chi_0\circ\psi+\nu$ on this set, we find a solution 
$f_\nu$ such that
$$\int_{M_{b_0}}|f_\nu|^2e^{-\chi_0\circ\psi}dV\le
\int_M|f_\nu|^2e^{-\chi\circ\psi}dV\le 
\int_{U\ssm M_b}|g|^2e^{-\chi\circ\psi}dV\le Ce^{-\nu},$$
thus $f_\nu$ converges to $0$ in $L^2(M_{b_0})$ and $h_\nu=\eta h-f_\nu
\in \cC^\infty_{n,q-1}(M,E)$ is a $D''$-closed form converging to $h$ 
in $L^2(M_{b_0})$. However, if we choose
the minimal solution such that $\delta''_\chi f_\nu=0$ as in Rem.~VIII-4.6,
we get $\Delta''_\chi f_\nu=\delta''_\chi g$ on $M$ and in particular
$\Delta''_{\chi_0}f_\nu=0$ on $M_{b_0}$. G\aa rding's inequality
VI-3.3 applied to the elliptic operator $\Delta''_{\chi_0}$
shows that $f_\nu$ converges to $0$ with all derivatives on $L$,
hence $h_\nu$ converges to $h$ on $L$. Now, replace $L$ by an
exhaustion $L_\nu$ of $U$ by compact sets; some diagonal subsequence 
$h_\nu$ converges to $h$ in $\cC^\infty_{n,q-1}(U,E)$.\qed
\endproof

\subsection{4.B}{A Local Vanishing Result for Sheaves}
Let $(X,\cO_X)$ be an analytic scheme and $\cS$ a coherent sheaf of
$\cO_X$-modules.  We wish to extend Prop.~4.1 to the cohomology
groups $H^k(X,\cS)$.  The first step is to show that the result holds on 
small open sets, and this is done by means of local resolutions of $\cS$. 

For a given point $x\in X$, we choose a patch $(A,\cO_\Omega/\cJ)$ of $X$
containing $x$, where $A$ is an analytic subset of $\Omega\subset\bC^N$
and $\cJ$ a sheaf of ideals with zero set $A$. Let $i_A:A\longrightarrow\Omega$ be 
the inclusion. Then $(i_A)_\star\cS$ is a coherent $\cO_\Omega$-module 
supported on $A$. In particular there is a neighborhood
$W_0\subset\Omega$ of $x$ and a surjective sheaf morphism 
$$\cO^{p_0}\longrightarrow(i_A)_\star\cS~~~\hbox{\rm on}~~W_0,~~~~(u_1,\ldots,u_{p_0})
\longmapsto\sum_{1\le j\le p_0}u_jG_j$$
where $G_1,\ldots,G_{p_0}\in\cS(A\cap W_0)$ are generators of $(i_A)_\star\cS$ 
on $W_0$. If we repeat the procedure inductively for the kernel of the above
surjective morphism, we get a {\it homological free resolution} of 
$(i_A)_\star\cS\,$:
$$\cO^{p_l}\longrightarrow\cdots\longrightarrow\cO^{p_1}\longrightarrow\cO^{p_0}\longrightarrow(i_A)_\star\cS\longrightarrow 0~~~
\hbox{\rm on}~~W_l\leqno(4.3)$$
of arbitrary large length $l$, on neighborhoods $W_l\subset
W_{l-1}\subset \ldots\subset W_0$.  In particular, after replacing $\Omega$ 
by $W_{2N}$ and $A$ by $A\cap W_{2N}$, we may assume that $(i_A)_\star\cS$
has a resolution of length $2N$ on $\Omega$. In this case, we shall say
that $A\subset\Omega$ is a $\cS$-{\it distinguished patch} of $X$.

\claim{(4.4) Lemma} Let $A\subset\Omega$ be a $\cS$-distinguished patch
of $X$ and $U$ a strongly $q$-convex open subset of $A$. Then
$$H^k(U,\cS)=0~~~~\hbox{\it for}~~k\ge q.$$
\endclaim

\proof{Proof} Theorem 2.13 shows that there exists a strongly $q$-convex
open set $V\subset\Omega$ such that $U=A\cap V$.  Let us denote by
$\cZ^l$ the kernel of $\cO^{p_l}\longrightarrow\cO^{p_{l-1}}$ for $l\ge 1$ and
$\cZ^0=\ker\big(\cO^{p_0}\longrightarrow(i_A)_\star\cS\big)$.  There are exact
sequences $$\eqalign{ &0\longrightarrow\cZ^0\longrightarrow\cO^{p_0}\longrightarrow(i_A)_\star\cS\longrightarrow 0,\cr
&0\longrightarrow\cZ^l\longrightarrow\cO^{p_l}\longrightarrow\cZ^{l-1}\longrightarrow 0,~~~~1\le l\le 2N.\cr}$$
For $k\ge q$, Prop.~4.1~a) gives $H^k(V,\cO^{p_l})=0$, 
therefore we get
$$H^k(U,\cS)\simeq H^k\big(V,(i_A)_\star\cS\big) \simeq
H^{k+1}(V,\cZ^0)\simeq\ldots\simeq H^{k+2N+1}(V,\cZ^{2N}),$$ 
and the last group vanishes because $\hbox{\rm topdim}\,V\le\dim_\bR
V=2N$.\qed
\endproof

\subsection{4.C}{Topological Structure on Spaces of Sections and on
Cohomology Groups}
Let $V\subset\Omega$ be a strongly $1$-complete open set
relatively to a $\cS$-distinguished patch $A\subset\Omega$
and let $U=A\cap V$.  By the proof of Lemma~4.4, we have 
$$H^1(V,\cZ^0)\simeq H^{2N+1}(V,\cZ^{2N})=0,$$
hence we get an exact sequence
$$0\longrightarrow\cZ^0(V)\longrightarrow\cO^{p_0}(V)\longrightarrow\cS(U)\longrightarrow 0.\leqno(4.5)$$
We are going to show that the Fr\'echet space structure on $\cO^{p_0}(V)$
induces a natural Fr\'echet space structure on the groups of sections of
$\cS$ over any open subset. We first note that $\cZ^0(V)$ is closed in
$\cO^{p_0}(V)$. Indeed, let $f_\nu\in\cZ^0(V)$ be a sequence converging 
to a limit $f\in\cO^{p_0}(V)$ uniformly on compact subsets of $V$.
For every $x\in V$, the germs $(f_\nu)_x$ converge to $f_x$ with
respect to the topology defined by (1.4) on $\cO^{p_0}$. As $\cZ^0_x$
is closed in $\cO_x^{p_0}$ in view of Th.~1.5~b), we get $f_x\in\cZ^0_x$
for all $x\in V$, thus $f\in\cZ^0(V)$. 

\claim{(4.6) Proposition} The quotient topology on $\cS(U)$ is 
independent of the choices made above.
\endclaim

\proof{Proof} For a smaller set $U'=A\cap V'$ where $V'$ is a strongly $1$-convex 
open subset of $V$, the restriction map $\cO^{p_0}(V)\longrightarrow\cO^{p_0}(V')$ 
is continuous, thus $\cS(U)\longrightarrow\cS(U')$ is continuous.  If $(V_\alpha)$ is a 
countable covering of $V$ by such sets and $U_\alpha=A\cap V_\alpha$, we 
get an injection of $\cS(U)$ onto the closed subspace of the product
$\prod\cS(U_\alpha)$ consisting of families which are compatible in the
intersections.  Therefore, the Fr\'echet topology induced by the product
coincides with the original topology of $\cS(U)$.  If we choose other
generators $H_1,\ldots,H_{q_0}$ for $(i_A)_\star\cS$, the germs $H_{j,x}$
can be expressed in terms of the $G_{j,x}\,$'s, thus we get a
commutative diagram 
$$\cmalign{ 
&\cO^{p_0}(V)&\buildo G\over\longrightarrow&\cS(U)&\longrightarrow 0\cr 
&~~~\big\downarrow&&~~\big|\big|&\cr
&\cO^{q_0}(V)&\buildo H\over\longrightarrow&\cS(U)&\longrightarrow 0\cr}$$ 
provided that $U$ and $V$ are small enough.  If we express the
generators $G_j$ in terms of the $H_j$'s, we find a similar diagram with
opposite vertical arrows and we conclude easily that the topology
obtained in both cases is the same.  Finally, it remains to show that
the topology of $\cS(U)$ is independent of the embedding
$A\subset\Omega$ near a given point $x\in X$.  We compare the given
embedding with the Zariski embedding $(A,x)\subset\Omega'$ of minimal
dimension $d$.  After shrinking $A$ and changing coordinates, we may
assume $\Omega=\Omega'\times\bC^{N-d}$ and that the embedding
$i_A:A\longrightarrow\Omega$ is the composite of $i'_A:A\longrightarrow\Omega'$ and of the 
inclusion $j:\Omega'\longrightarrow\Omega'\times\{0\}\subset\Omega$. For
$V'\subset\Omega'$ sufficient small and $U'=A\cap V'$, we have a
surjective map $G':\cO^{p_0}(V')\longrightarrow\cS(U')$ obtained by choosing generators
$G'_j$ of $(i'_A)^\star\cS$ on a neighborhood of $x$ in $\Omega'$. 
Then we consider the open set $V=V'\times\bC^{N-d}\subset\Omega$
and the surjective map onto $\cS(U')$ equal to the composite 
$$\cO^{p_0}(V)\buildo j^\star\over\longrightarrow
\cO^{p_0}(V')\buildo G'\over\longrightarrow\cS(U).$$ 
This map corresponds to a choice of generators $G_j\in(i_A)^\star\cS(V)$ 
equal to the functions $G'_j$, considered as functions independent of 
the last variables $z_{d+1},\ldots,z_N$.  Since $j^\star$ is open, it is
obvious that the quotient topology on $\cS(U')$ is the same for both
embeddings.\qed
\endproof

Now, there is a natural topology on the cohomology groups $H^k(X,\cS)$.
In fact, let $(U_\alpha)$ be a countable covering of $X$
by strongly \hbox{$1$-complete} open sets, such that each
$U_\alpha$ is contained in a $\cS$-distinguished patch.  Since the
intersections $U_{\alpha_0\ldots\alpha_k}$ are again strongly
1-complete, the covering $\cU$ is acyclic by Lemma~4.4 
and Leray's theorem shows that $H^k(X,\cS)$ is isomorphic to 
$\check H^q(\cU,\cS)$. 
We consider the product topology on the spaces of \v Cech cochains
$C^k(\cU,\cS)=\prod\cS(U_{\alpha_0\ldots\alpha_k})$ and the quotient
topology on $\check H^k(\cU,\cS)$.  It is clear that $\check
H^0(\cU,\cS)$ is a Fr\'echet space; however the higher cohomology
groups $\check H^k(\cU,\cS)$ need not be Hausdorff because the
coboundary groups may be non closed in the cocycle groups.  
The resulting topology on $H^k(X,\cS)$ is independent of the choice 
of the covering: in fact we only have to check that the bijective
continuous map $\check H^k(\cU,\cS)\longrightarrow\check H^k(\cU',\cS)$ 
is a topological isomorphism if $\cU'$ is a refinement of
$\cU$, and this follows from Rem.~1.10 applied to the morphism 
of \v Cech complexes $C^\bu(\cU,\cS)\longrightarrow C^\bu(\cU',\cS)$.

Finally, observe that when $\cS$ is the locally free sheaf associated
to a holomorphic vector bundle $E$ on a smooth manifold $X$, the topology 
on $H^k\big(X,\cO(E)\big)$ is the same as the topology associated to the
Fr\'echet space structure on the Dolbeault complex 
$\big(\cC^\infty_{0,\bu}(X,E),d''\big)\,$: by the analogue of formula (IV-6.11) 
we have a bijective continuous map
$$\eqalign{
\check H^k\big(\cU,\cO(E)\big)&\longrightarrow H^k\big(\cC^\infty_{0,\bu}(X,E)\big)\cr
\hfill \{(c_{\alpha_0\ldots\alpha_k})\}&\longmapsto
f(z)=\sum_{\alpha_0,\ldots,\alpha_q}c_{\alpha_0\ldots\alpha_q}(z)\,\theta_{\alpha_q}
\,d''\theta_{\alpha_0}\wedge\ldots\wedge d''\theta_{\alpha_{q-1}}\cr}$$
where $(\theta_\alpha)$ is a partition of unity subordinate to $\cU$.
As in Rem.~1.10, the continuity of the inverse follows by the open 
mapping theorem applied to the surjective map
$$Z^k\big(C^\bu(\cU,\cO(E))\big)\oplus \cC^\infty_{0,k-1}(X,E)
\longrightarrow Z^k\big(\cC^\infty_{0,\bu}(X,E)\big).$$
We shall need a few simple additional results.

\claim{(4.7) Proposition} The following properties hold:
\smallskip
\item{\rm a)} For every $x\in X$, the map $\cS(X)\longrightarrow\cS_x$ is 
continuous with respect to the topology of $\cS_x$ defined by $(1.4)$.
\smallskip
\item{\rm b)} If $\cS'$ is a coherent analytic subsheaf of $\cS$, 
the space of global sections $\cS'(X)$ is closed in $\cS(X)$.
\smallskip
\item{\rm c)} If $U'\subset U$ are open in $X$,
the restriction maps $H^k(U,\cS)\longrightarrow H^k(U',\cS)$ are continuous.
\smallskip
\item{\rm d)} If $U'$ is relatively compact in $U$, the
restriction operator $\cS(U)\longrightarrow\cS(U')$ is compact.
\smallskip
\item{\rm e)} Let $\cS\longrightarrow\cS'$ be a morphism of coherent sheaves
over $X$. Then the induced maps $H^k(X,\cS)\longrightarrow H^k(X,\cS')$ are 
continuous.\smallskip
\endclaim

\proof{Proof} a) Let $V\subset\Omega$ be a strongly $1$-convex open
neighborhood of $x$ relatively to a $\cS$-distinguished patch
$A\subset\Omega$. The map $\cO^{p_0}(V)\longrightarrow\cO^{p_0}_x$ is continuous, 
and the same is true for $\cO^{p_0}_x\longrightarrow\cS_x$ by \S 1. Therefore 
the composite $\cO^{p_0}(V)\longrightarrow\cS_x$ and its factorization $\cS(U)\longrightarrow\cS_x$ 
are continuous.
\medskip
\noindent{b)} is a consequence of the above property a) and of the fact 
that each stalk $\cS'_x$ is closed in $\cS_x$ (cf.\ 1.5 b)).
\medskip
\noindent{c)} The restriction map $\cS(U)\longrightarrow\cS(U')$ is continuous, and the
case of higher cohomology groups follows immediately.
\medskip
\noindent{d)} Assume first that $U=A\cap V$ and $U'=A\cap V'$, where $A\subset
\Omega$ is a \hbox{$\cS$-distinguished} patch and $V'\compact V$ are strongly 
$1$-convex open subsets of $\Omega$.  The operator
$\cO^{p_0}(V)\longrightarrow\cO^{p_0}(V')$ is compact by Montel's theorem, thus
$\cS(U)\longrightarrow\cS(U')$ is also compact. In the general case, select
a finite family of strongly $1$-convex sets 
$U'_\alpha\compact U_\alpha\subset U$ such that $(U'_\alpha)$
covers $\ol U'$ and $U_\alpha$ is contained in some distinguished patch.
There is a commutative diagram
$$\cmalign{
\noalign{\vskip2pt}
&~~~\cS(U)~~\rightarrowfil~~&\cS(U')\cr
\noalign{\vskip4pt}
&~~~~~~\big\downarrow&~~~\big\downarrow\cr
\noalign{\vskip4pt}
&\prod\cS(U_\alpha)\longrightarrow\prod\cS(U'_\alpha)\longrightarrow\prod\cS&(U'\cap U'_\alpha)\cr}$$
where the right vertical arrow is a monomorphism and where the first arrow in 
the bottom line is compact. Thus $\cS(U)\longrightarrow\cS(U')$ is compact.
\medskip
\noindent{e)} It is enough to check that $\cS(U)\longrightarrow\cS'(U)$ is continuous, and
for this we may assume that $U=A\cap V$ where $V$ is a small neighborhood of
a given point $x$. Let $G_1,\ldots,G_{p_0}$ be generators of $\cS_x$,
$G'_1,\ldots,G'_{p_0}$ their images in $\cS'_x$. Complete these elements
in order to obtain a system of generators $(G'_1,\ldots,G'_{q_0})$ of $\cS'_x$.
For $V$ small enough, the map $\cS(U)\longrightarrow\cS'(U)$ is induced by the
inclusion $\cO^{p_0}(V)\longrightarrow\cO^{p_0}(V)\times\{0\}\subset\cO^{q_0}(V)$,
hence continuous.\qed
\endproof

\subsection{4.D}{Cartan-Serre Finiteness Theorem}
The above results enable us to prove a finiteness theorem for 
cohomology groups over compact analytic schemes.

\claim{(4.8) Theorem {\rm(Cartan-Serre)}} Let $\cS$ be a coherent 
analytic sheaf over an analytic scheme $(X,\cO_X)$. If $X$ is compact, all
cohomology groups $H^k(X,\cS)$ are finite dimensional $($and Hausdorff~$)$.
\endclaim

\proof{Proof} There exist finitely many strongly $1$-complete 
open sets $U'_\alpha\compact U_\alpha$ such that each 
$U_\alpha$ is contained in some $\cS$-distinguished patch and such that 
$\bigcup U'_\alpha=X$.  By Prop.~4.7~d), the restriction map 
on \v Cech cochains
$$C^\bu(\cU,\cS)\longrightarrow C^\bu(\cU',\cS)$$
defines a compact morphism of complexes of Fr\'echet spaces.
As the coverings $\cU=(U_\alpha)$ and $\cU'=(U'_\alpha)$ are acyclic 
by 4.4, the induced map 
$$\check H^k(\cU,\cS)\longrightarrow\check H^k(\cU',\cS)$$
is an isomorphism, both spaces being isomorphic to $H^k(X,\cS)$.
We conclude by Schwartz' theorem 1.9.\qed
\endproof

\subsection{4.E}{Local Approximation Theorem}
We show that a local analogue of the approximation result 4.1 b) holds
for a sheaf $\cS$ over an analytic scheme $(X,\cO_X)$.

\claim{(4.9) Lemma} Let $A\subset\Omega$ be a $\cS$-distinguished patch of
$X$, and $U'\subset U\subset A$ open subsets such that
$U'$ is $q$-Runge complete in $U$. Then the restriction map
$$H^{q-1}(U,\cS)\longrightarrow H^{q-1}(U',\cS)$$
has a dense range.
\endclaim

\proof{Proof} Let $L$ be an arbitrary compact subset of $U'$.
Proposition 2.16 applied with $Y=U$ embedded in some neighborhood in
$\Omega$ shows that there is a neighborhood $V$ of $U$ in $\Omega$ such
that $A\cap V=U$ and a strongly $q$-convex function $\psi$ on $V$ such that 
$L\subset U_b\compact U'$ for some $U_b=A\cap V_b$. The proof of 
Lemma~4.4 gives $H^q(V,\cZ^0)=H^q(V_b,\cZ^0)=0$ and the cohomology
exact sequences of $0\to\cZ^0\to\cO^{p_0}\to i_A^\star\cS\to 0$ 
over $V$ and $V_b$ yield
a commutative diagram of continuous maps
$$\cmalign{
&H^{q-1}\big(V,\cO^{p_0}\big)&~\longrightarrow~&H^{q-1}\big(V,i_A^\star\cS\big)&=
H^{q-1}\big(&U,\cS)\cr
\noalign{\vskip4pt}
&~~~~~~~~\big\downarrow&&~~~~~~~~\big\downarrow&&\big\downarrow\cr
\noalign{\vskip4pt}
&H^{q-1}\big(V_b,\cO^{p_0}\big)&~\longrightarrow~&H^{q-1}\big(V_b,i_A^\star\cS\big)&=
H^{q-1}\big(&U_b,\cS)\cr}$$
where the horizontal arrows are surjective. Since $V_b$ is $q$-Runge
complete in $V$, the left vertical arrow has a dense range 
by Prop.~4.1 b). As $U'$ is the union of an increasing sequence
of sets $U_{b_\nu}$, we only have to show that the range remains dense
in the inverse limit $H^{q-1}(U',\cS)$. For that, we apply
Property 1.11~d) on a suitable covering of $U$.
Let $\cW$ be a countable basis of the topology of $U$, consisting of
strongly $1$-convex open subsets contained in $\cS$-distinguished 
patches. We let $\cW'$ (resp.\ $\cW_\nu$) be the subfamily of 
$W\in\cW$ such that $W\compact U'$ (resp.\ $W\compact U_{b_\nu}$). Then
$\cW,~\cW',~\cW_\nu$ are acyclic coverings of $U,~U',~U_{b_\nu}$
and each restriction map $C^\bu(\cW,\cS)\longrightarrow C^\bu(\cW_\nu,\cS)$
is surjective. Property 1.11~d) can thus be applied and the
lemma follows.\qed
\endproof

\subsection{4.F}{Statement and Proof of the Andreotti-Grauert Theorem}
\claim{(4.10) Theorem {\rm(\cite{Andreotti-Grauert~1962})}} Let $\cS$ be a 
coherent analytic sheaf over a strongly $q$-convex analytic scheme 
$(X,\cO_X)$. Then
\smallskip
\item{\rm a)} $H^k(X,\cS)$ is Hausdorff and finite dimensional
for $k\ge q$.
\smallskip
\noindent{}Moreover, let $U$ be a $q$-Runge open subset of $X$, $q\ge 1$. Then
\smallskip
\item{\rm b)} the restriction map $H^k(X,\cS)\to H^k(U,\cS)$ 
is an \hbox{isomorphism for $k\ge q\,;$}
\smallskip
\item{\rm c)} the restriction map $H^{q-1}(X,\cS)\to
H^{q-1}(U,\cS)$ has a dense range.
\endclaim

The compact case $q=0$ of 4.10~a) is precisely the Cartan-Serre
finiteness theorem.  For $q\ge 1$, the special case when $X$ is
strongly $q$-complete and $U=\emptyset$ yields the following very
important consequence. 

\claim{(4.11) Corollary} If $X$ is strongly $q$-complete, then 
$$H^k(X,\cS)=0~~~~\hbox{\it for}~~k\ge q.$$
\endclaim

Assume that $q\ge 1$ and let $\psi$ be a smooth exhaustion on $X$
that is strongly $q$-convex on $X\ssm K$. We first consider 
sublevel sets $X_d\supset X_c\supset K$, $d>c$,
and verify assertions 4.10 b), c) for all restriction maps 
$$H^k(X_d,\cS)\longrightarrow H^k(X_c,\cS),~~~~k\ge q-1.$$
The basic idea, already contained in \cite{Andreotti-Grauert~1962}, is 
to deform $X_c$ into
$X_d$ through a sequence of strongly $q$-convex open sets $(G_j)$
such that $G_{j+1}$ is obtained from $G_j$ by making a small bump.

\claim{(4.12) Lemma} There exist a sequence of strongly $q$-convex
open sets $G_0\subset\ldots\subset G_s$ and a sequence
of strongly $q$-complete open sets $U_0,\ldots,U_{s-1}$ in $X$ such that 
\smallskip
\item{\rm a)} $G_0=X_c$,~~~$G_s=X_d$,~~~$G_{j+1}=G_j\cup U_j$~~for~
$0\le j\le s-1\,;$
\smallskip
\item{\rm b)} $G_j=\{x\in X\,;\,\psi_j(x)<c_j\}$ where $\psi_j$ is an
exhaustion function on $X$ that is strongly $q$-convex on $X\ssm K\,;$
\smallskip
\item{\rm c)} $U_j$ is contained in a
$\cS$-distinguished patch $A_j\subset\Omega_j$ of $X\,;$
\smallskip
\item{\rm d)} $G_j\cap U_j$ is strongly $q$-complete and
$q$-Runge complete in $U_j$.
\endclaim

\proof{Proof} There exists a finite covering of the compact set 
$\ol X_d\ssm X_c$ by
$\cS$-distinguished patches $A_j\subset\Omega_j$, $0\le j<s$, where
$\Omega_j\subset\bC^{N_j}$ is a euclidean ball and $K\cap A_j=\emptyset$. 
Let $\theta_j\in\cD(X)$ be a family of functions such that 
$\Supp\theta_j\subset A_j$, $\theta_j\ge 0$,
$\sum\theta_j\le 1$ and $\sum\theta_j=1$ on a neighborhood of 
$\ol X_d\ssm X_c$. We can find $\varepsilon_0>0$ so small that 
$$\psi_j=\psi-\varepsilon\sum_{0\le k<j}\theta_k$$
is still strongly $q$-convex on $X\ssm K$ for $0\le j\le s$
and $\varepsilon\le\varepsilon_0$.  We have
$\psi_0=\psi$ and $\psi_s=\psi-\varepsilon$ on $\ol X_d\ssm X_c$, thus
$$G_j=\{x\in X\,;\,\psi_j(x)<c\},~~~~0\le j\le s$$
is an increasing sequence of strongly $q$-convex open sets such that
$G_0=X_c$, $G_s=X_{c+\varepsilon}$. Moreover, as $\psi_{j+1}-\psi_j=
-\varepsilon\theta_j$ has support in $A_j$, we have
$$G_{j+1}=G_j\cup U_j~~~~\hbox{\rm where}~~U_j=G_{j+1}\cap A_j.$$
It follows that conditions a), b), c) are satisfied with $c+\varepsilon$
instead of $d$. Finally, the functions
$$\varphi_j=1/(c-\psi_{j+1})+1/(r_j^2-|z-z_j|^2),~~~~
\wt\varphi_j=1/(c-\psi_j)+1/(r_j^2-|z-z_j|^2)$$
are strongly $q$-convex exhaustions on $U_j$ and $G_j\cap U_j=G_j\cap A_j$.
Let $L$ be an arbitrary compact subset of $G_j\cap U_j$ and
$a=\sup_L\psi_j<c$. Select $b\in]a,c[$ and set
$$\psi_{j,\eta}=\psi_j+\eta\varphi_j~~~~\hbox{\rm on}~~U_j,~~~~\eta>0.$$
Then $\psi_{j,\eta}$ is an exhaustion of $U_j$. As $\varphi_j$ is 
bounded below, we have
$$L\subset\{\psi_{j,\eta}<b\}\compact\{\psi_j<c\}\cap U_j=G_j\cap U_j$$
for $\eta$ small enough. Moreover
$$(1-\alpha)\psi_j+\alpha\psi_{j+1}=\psi-\varepsilon\sum_{0\le k<j}
\theta_k-\alpha\varepsilon\,\theta_j$$ 
is strongly $q$-convex for all $\alpha\in[0,1]$ and $\varepsilon\le
\varepsilon_0$ small enough, so Prop.~2.4 implies that $\psi_{j,\eta}$
is strongly $q$-convex. By definition, $G_j\cap U_j$ is thus 
\hbox{$q$-Runge} complete in $U_j$, and Lemma~4.12 
is proved with $X_{c+\varepsilon}$ instead of $X_d$.  In order to 
achieve the proof, we consider an increasing sequence 
$c=c_0<c_1<\ldots<c_N=d$ with $c_{k+1}-c_k\le\varepsilon_0$ and perform
the same construction for each pair $X_{c_k}\subset X_{c_{k+1}}$, with
$c$ replaced by $c_k$ and $\varepsilon=c_{k+1}-c_k$.\qed
\endproof

\claim{(4.13) Proposition} For every sublevel set $X_c\supset K$, the
group $H^k(X_c,\cS)$ is Hausdorff and finite dimensional when $k\ge q$.
Moreover, for $d>c$, the restriction map
$$H^k(X_d,\cS)\longrightarrow H^k(X_c,\cS)$$
is an isomorphism when $k\ge q$ and has a dense range when $k=q-1$.
\endclaim

\proof{Proof} Thanks to Lemma~4.12, we are led to consider the restriction maps
$$H^k(G_{j+1},\cS)\longrightarrow H^k(G_j,\cS).\leqno(4.14)$$
Let us apply the Mayer-Vietoris exact sequence IV-3.11 to $G_{j+1}=G_j\cup U_j$.
For $k\ge q$ we have $H^k(U_j,\cS)=H^k(G_j\cap U_j,\cS)=0$ by Lemma~4.4.
Hence we get an exact sequence
$$\cmalign{
H^{q-1}(G_{j+1},\cS)&\longrightarrow H^{q-1}(G_j,\cS)\oplus H^{q-1}(U_j,\cS)&\longrightarrow 
H^{q-1}(G_j\cap U_j,\cS)\longrightarrow\cr
H^k(G_{j+1},\cS)&\longrightarrow~~~~~~~~~~~~~H^k(G_j,\cS)&\longrightarrow~~~0~~~\longrightarrow\cdots,~~~~
k\ge q.\cr}$$
In this sequence, all the arrows are induced by restriction maps, so
they define continuous linear operators. We already infer that
the map (4.14) is bijective for $k>q$ and surjective for $k=q$.
There exist a $\cS$-acyclic covering $\cV=(V_\alpha)$ of $X_d$ and 
a finite family $\cV'=(V'_{\alpha_1},\ldots,V'_{\alpha_p})$ of open sets 
such that $V'_{\alpha_j}\compact V_{\alpha_j}$ and $\bigcup V'_{\alpha_j}
\supset\smash{\ol X_c}$. Let $\cW$ be a locally finite $\cS$-acyclic covering 
of $X_c$ which refines $\cV'\cap X_c=(V'_{\alpha_j}\cap X_c)$.
The refinement map
$$C^\bu(\cV,\cS)\longrightarrow C^\bu(\cV'\cap X_c,\cS)\longrightarrow C^\bu(\cW,\cS)$$
is compact because the first arrow is, and it induces a surjective map
$$H^k(X_d,\cS)\longrightarrow H^k(X_c,\cS)~~~~\hbox{\rm for}~~k\ge q.$$
By Schwartz' theorem 1.9, we conclude that $H^k(X_c,\cS)$ is Hausdorff and 
finite dimensional for $k\ge q$. This is equally true for $H^q(G_j,\cS)$ 
because $G_j$ is also a global sublevel set $\{x\in X\,;\,\psi_j(x)<c_j\}$
containing $K$. Now, the Mayer-Vietoris exact sequence implies that the
composite
$$H^{q-1}(U_j,\cS)\longrightarrow H^{q-1}(G_j\cap U_j,\cS)\buildo\partial\over
\longrightarrow H^q(G_{j+1},\cS)$$
is equal to zero. However, the first arrow has a dense range by
Lemma~4.9. As the target space is Hausdorff, the second arrow must
be zero; we obtain therefore the injectivity of
$H^q(G_{j+1},\cS)\longrightarrow H^q(G_j,\cS)$ and an exact sequence
$$\cmalign{
H^{q-1}(G_{j+1},\cS)\longrightarrow H^{q-1}(G_j,\cS)&\oplus H^{q-1}(U_j,\cS)
&\longrightarrow H^{q-1}(G_j\cap U_j,\cS)\longrightarrow 0\cr
\hfill g&\oplus u&\longmapsto u_{\restriction G_j\cap U_j}-
g_{\restriction G_j\cap U_j}.\cr}$$
The argument used in Rem.~1.10 shows that the surjective arrow is open.
Let $g\in H^{q-1}(G_j,\cS)$ be given. By Lemma~4.9, we can
approximate $g_{\restriction G_j\cap U_j}$ by a sequence
$u_{\nu\restriction G_j\cap U_j}$, $u_\nu\in H^{q-1}(U_j,\cS)$.
Then $w_\nu=u_{\nu\restriction G_j\cap U_j}-g_{\restriction G_j\cap U_j}$
tends to zero. As the second map in the exact sequence is open, we can
find a sequence
$$g'_\nu\oplus u'_\nu\in H^{q-1}(G_j,\cS)\oplus H^{q-1}(U_j,\cS)$$
converging to zero which is mapped on $w_\nu$. Then
$(g-g'_\nu)\oplus(u_\nu-u'_\nu)$ is mapped on zero, and there exists
a sequence $f_\nu\in H^{q-1}(G_{j+1},\cS)$ which coincides
with $g-g'_\nu$ on $G_j$ and with $u_\nu-u'_\nu$ on $U_j$. In particular
$f_{\nu\restriction G_j}$ converges to $g$ and we have shown that
$$H^{q-1}(G_{j+1},\cS)\longrightarrow H^{q-1}(G_j,\cS)$$
has a dense range.\qed
\endproof

\proof{Proof of Andreotti-Grauert's Theorem 4.10.} Let $\cW$
be a countable basis of the topology of $X$
consisting of strongly $1$-convex open sets $W_\alpha$
contained in $\cS$-distinguished patches of $X$. 
Let $L\subset U$ be an arbitrary compact subset. Select
a smooth exhaustion function $\psi$ on $X$ such that $\psi$ is
strongly $q$-convex on $X\ssm\ol X_b$ and
$L\subset X_b\compact U$ for some sublevel set $X_b$ of $\psi\,$;
choose $c>b$ such that $X_c\compact U$.
For every $d\in\bR$, we denote by $\cW_d\subset\cW$
the collection of sets $W_\alpha\in\cW$ such that $W_\alpha\subset X_d$.
Then $\cW_d$ is a $\cS$-acyclic covering of $X_d$. We consider the 
sequence of \v Cech complexes
$$E^\bu_\nu=C^\bu(\cW_{c+\nu},\cS),~~~~\nu\in\bN$$
together with the surjective projection maps $E^\bu_{\nu+1}\longrightarrow E^\bu_\nu$, 
and their inverse limit $E^\bu=C^\bu(\cW,\cS)$. Then we have
$H^k(E^\bu)=H^k(X,\cS)$ and $H^k(E^\bu_\nu)=H^k(X_{c+\nu},\cS)$.
Propositions 1.11 (a,b,c) and 4.13 imply that $H^k(X,\cS)\longrightarrow
H^k(X_c,\cS)$ is bijective for $k\ge q$ and has a dense range for
$k=q-1$. It already follows that $H^k(X,\cS)$ is Hausdorff for
$k\ge q$. Now, take an increasing sequence of open sets
$X_{c_\nu}$ equal to sublevel sets of a sequence of exhaustions 
$\psi_\nu$, such that $U=\bigcup X_{c_\nu}$. Then all groups
$H^k(X_{c_\nu},\cS)$ are in bijection with $H^k(X,\cS)$ for $k\ge q$,
and the image of $H^{q-1}(X_{c_{\nu+1}},\cS)$ in $H^{q-1}(X_{c_\nu},\cS)$
is dense because it contains the image of $H^{q-1}(X,\cS)$.
Proposition 1.11 (a,b,c) again shows that 
$H^k(U,\cS)\longrightarrow H^k(X_{c_0},\cS)$ is bijective
for $k\ge q$, and d) shows that $H^{q-1}(X,\cS)\longrightarrow H^{q-1}(U,\cS)$
has a dense range. The theorem follows.\qed
\endproof

A combination of Andreotti-Grauert's theorem with Th.~3.6 
yields the following important consequence.

\claim{(4.15) Corollary} Let $\cS$ be a coherent sheaf over an analytic 
scheme $(X,\cO_X)$ with $\dim X\le n$.
\smallskip
\item{\rm a)} We have~~$H^k(X,\cS)=0$~~for all $k\ge n+1\,;$
\smallskip
\item{\rm b)} If $X$ has no $n$-dimensional compact irreducible component,
then we have\break \hbox{$H^n(X,\cS)=0$}.
\smallskip
\item{\rm c)} If $X$ has only finitely many $n$-dimensional compact
irreducible components, then $H^n(X,\cS)$ is finite dimensional.
\qed\smallskip
\endclaim

The special case of 4.15 b) when $X$ is smooth and $\cS$ locally free
has been first proved by \cite{Malgrange~1955}, and the general case is
due to \cite{Siu~1969}. Another consequence is the following
approximation theorem for coherent sheaves over manifolds, which
results from Prop.~3.8.

\claim{(4.16) Proposition} Let $\cS$ be a coherent sheaf 
over a non compact connected complex manifold $M$ with $\dim M=n$.
Let $U\subset M$ be an open subset such that the complement
$M\ssm U$ has no compact connected component. Then the
restriction map $H^{n-1}(M,\cS)\to H^{n-1}(U,\cS)$ has a dense range.\qed
\endclaim

\section{5}{Grauert's Direct Image Theorem}
The goal of this section is to prove the following fundamental
result on direct images of coherent analytic sheaves, due to
\cite{Grauert~1960}.

\claim{(5.1) Direct image theorem} Let $X$, $Y$ be complex
analytic schemes and let $F:X\to Y$ be a proper analytic morphism.
If $\cS$ is a coherent $\cO_X$-module, the direct
images $R^qF_\star\cS$ are coherent $\cO_Y$-modules.
\endclaim

We give below a beautiful proof due to \cite{Kiehl-Verdier~1971}, which 
is much simpler than Grauert's original proof; this proof rests
on rather deep properties of nuclear modules over nuclear Fr\'echet
algebras. We first introduce the basic concept of topological tensor 
product. Our presentation owes much to the seminar lectures by
\cite{Douady-Verdier~1973}.

\subsection{5.A}{Topological Tensor Products and Nuclear Spaces}
The algebra of holomorphic functions on a product space $X\times Y$
is a completion $\cO(X)\hotimes\cO(Y)$ of the algebraic tensor
product $\cO(X)\otimes\cO(Y)$. We are going to describe the
construction and the basic properties of the required topological 
tensor products $\hotimes$.

Let $E$, $F$ be (real or complex) vector spaces equipped with semi-norms
$p$ and $q$, respectively. Then $E\otimes F$ can be equipped with any
one of the two natural semi-norms $p\otimes_\pi q$, 
$p\otimes_\varepsilon q$ defined by
$$\eqalign{
p\otimes_\pi q(t)&=\inf\Big\{\sum_{1\le j\le N}p(x_j)\,q(y_j)\,;~
t=\!\!\sum_{1\le j\le N}x_j\otimes y_j\,,~x_j\in E\,,~y_j\in F~\Big\},\cr
p\otimes_\varepsilon q(t)&=\sup_{||\xi||_p\le 1,\,||\eta||_q\le 1}
\big|\xi\otimes\eta(t)\big|,~~~~\xi\in E',~~\eta\in F'\,;\cr}$$
the inequalities in the last line mean that $\xi$, $\eta$ satisfy $|\xi(x)|
\le p(x)$ and $|\eta(y)|\le q(y)$ for all $x\in E$, $y\in F$. Then clearly
$p\otimes_\varepsilon q\le p\otimes_\pi q$, for
$$p\otimes_\varepsilon q\Big(\sum x_j\otimes y_j\Big)\le
\sum p\otimes_\varepsilon q(x_j\otimes y_j)\le\sum p(x_j)\,q(y_j).$$
Given $x\in E$, $y\in F$, the Hahn-Banach theorem implies that there exist
$\xi$,~$\eta$ such that $||\xi||_p=||\eta||_q=1$ with
$\xi(x)=p(x)$ and $\eta(y)=q(y)$, hence 
$p\otimes_\varepsilon q(x\otimes y)\ge p(x)\,q(y)$. On the other hand
$p\otimes_\pi q(x\otimes y)\le p(x)\,q(y)$, thus
$$p\otimes_\varepsilon q(x\otimes y)=p\otimes_\pi q(x\otimes y)=p(x)\,q(y).$$

\claim{(5.2) Definition} Let $E$, $F$ be locally convex topological vector
spaces. The topological tensor product $E\hotimes_\pi F$ $($resp.\
$E\hotimes_\varepsilon F\,)$ is the Hausdorff completion of $E\otimes F$,
equipped with the family of semi-norms $p\otimes_\pi q$ $($resp.\ 
$p\otimes_\varepsilon q)$ associated to fundamental families of 
semi-norms on $E$ and $F$.
\endclaim

Since we may also write
$$p\otimes_\pi q(t)=\inf\Big\{\sum|\lambda_j|\,;~
t=\sum\lambda_j\,x_j\otimes y_j\,,~p(x_j)\le 1\,,~q(y_j)\le 1~\Big\}$$
where the $\lambda_j$'s are scalars, we see that the closed unit ball
$B(p\hotimes_\pi q)$ in $E\hotimes_\pi F$ is the closed convex hull
of $B(p)\otimes B(q)$. From this, we easily infer that the topological
dual space $(E\hotimes_\pi F)'$ is isomorphic to the space of continuous
bilinear forms on $E\times F$. Another simple consequence of this 
interpretation of $B(p\hotimes_\pi q)$ is example a) below.

\claim{\bf(5.3) Examples} \smallskip\rm
\noindent{a)} For all discrete spaces $I$, $J$, there is an isometry 
$$\ell^1(I)\hotimes_\pi\ell^1(J)\simeq\ell^1(I\times J).$$
\noindent{b)} For Banach spaces $(E,p)$, $(F,q)$, the closed unit
ball in $E\hotimes_\varepsilon F$ is dual to the unit ball
$B(p'\hotimes_\pi q')$ of $E'\hotimes_\pi F'$ through the natural
pairing extending the algebraic pairing of $E\otimes F$ and $E'\otimes F'$.
If $c_0(I)$ denotes the space of bounded sequences on $I$ converging to 
zero at infinity, we have $c_0(I)'=\ell^1(I)$, hence
by duality $c_0(I)\hotimes_\varepsilon c_0(J)$ is isometric to
$c_0(I\times J)$.
\medskip
\noindent{c)} If $X$, $Y$ are compact topological spaces and if $C(X)$, $C(Y)$
are their algebras of continuous functions with the sup norm, then
$$C(X)\hotimes_\varepsilon C(Y)\simeq C(X\times Y).$$
Indeed, $C(X)'$ is the space of finite Borel measures equipped with the 
mass norm. Thus for $f\in C(X)\otimes C(Y)$, the $\otimes_\varepsilon$-seminorm
is given by
$$||f||_\varepsilon=\sup_{||\mu||\le 1,\,||\nu||\le 1}\mu\otimes\nu(f)=
\sup_{X\times Y}|f|\,;$$
the last equality is obtained by taking Dirac measures $\delta_x$, $\delta_y$
for $\mu$, $\nu$ (the inequality $\le$ is obvious). Now $C(X)\otimes C(Y)$ is 
dense in $C(X\times Y)$ by the Stone-Weierstrass theorem, hence its completion
is $C(X\times Y)$, as desired.\qed
\endclaim

Let $f:E_1\to E_2$ and $g:F_1\to F_2$ be continuous morphisms. For all
semi-norms $p_2$, $q_2$ on $E_2$, $F_2$, there exist semi-norms
$p_1$, $q_1$ on $E_1$, $F_1$ and constants $||f||=||f||_{p_1,p_2}$,
$||g||=||g||_{q_1,q_2}$ such that
$p_2\circ f\le ||f||\,p_1$ and $q_2\circ g\le ||g||\,q_1$. Then we find
$$(p_2\otimes_\pi q_2)\circ(f\otimes g)\le ||f||\,||g||\,p_1\otimes_\pi q_1$$
and a similar formula with $p_j\otimes_\varepsilon q_j$. It follows that
there are well defined continuous maps
\medskip\noindent
$\cmalign{
(5.4')\hfill\qquad
&f\hotimes_\pi g&{}:E_1\hotimes_\pi F_1&{}\longrightarrow E_2\hotimes_\pi F_2,\cr
(5.4'')\hfill\qquad
&f\hotimes_\varepsilon g&{}:E_1\hotimes_\varepsilon F_1&{}\longrightarrow
E_2\hotimes_\varepsilon F_2.\cr}$
\medskip\noindent
Another simple fact is that $\hotimes_\pi$ preserves open morphisms:

\claim{(5.5) Proposition} If $f:E_1\to E_2$ and $g:F_1\to F_2$ are
epimorphisms, then $f\hotimes_\pi g:E_1\hotimes_\pi F_1\longrightarrow 
E_2\hotimes_\pi F_2$ is an epimorphism.
\endclaim

\proof{Proof} Recall that when $E$ is locally convex complete and $F$ Hausdorff,
a morphism $u:E\to F$ is open if and only if $\ol{u(V)}$ is a neighborhood
of $0$ for every neighborhood of $0$ (this can be checked essentially by the
same proof as 1.8~b)). Here, for any semi-norms $p$, $q$ on
$E_1$, $F_1$ the closure of $f\hotimes_\pi g\big(B(p\hotimes_\pi q)\big)$
contains the closed convex hull of $f\big(B(p)\big)\otimes g\big(B(q)\big)$
in which $f\big(B(p)\big)$ and $g\big(B(q)\big)$ are neighborhoods
of~$0$, so it is a neighborhood of $0$ in $E\hotimes_\pi F$.\qed
\endproof

If $E_1\subset E_2$ is a closed subspace, every continuous semi-norm
$p_1$ on $E_1$ is the restriction of a continuous semi-norm on $E_2$, and
every linear form $\xi_1\in E_1'$ such that $||\xi_1||_{p_1}\le 1$ can
be extended to a linear form $\xi_2\in E_2$ such that
$||\xi_2||_{p_2}=||\xi_1||_{p_1}$ (Hahn-Banach theorem); similar
properties hold for a closed subspace $F_1\subset F_2$. We infer that
$$(p_2\otimes_\varepsilon q_2)_{\restriction E_1\otimes F_1}=
p_1\otimes_\varepsilon q_1\,,$$
thus $E_1\hotimes_\varepsilon F_1$ is a closed subspace of
$E_2\hotimes_\varepsilon F_2$. In other words:

\claim{(5.6) Proposition} If $f:E_1\to E_2$ and $g:F_1\to F_2$ are
monomorphisms, then $f\hotimes_\varepsilon g:E_1\hotimes_\varepsilon F_1\longrightarrow 
E_2\hotimes_\varepsilon F_2$ is a monomorphism.\qed
\endclaim

Unfortunately, 5.5 fails for $\hotimes_\varepsilon$ and 5.6 fails for
$\hotimes_\pi$, even with Fr\'echet or Banach spaces. It follows that 
neither $\hotimes_\pi$ nor $\hotimes_\varepsilon$ are exact functors 
in the category of Fr\'echet spaces. In order to circumvent this difficulty, 
it is necessary to work in a suitable subcategory.

\claim{(5.7) Definition} A morphism $f:E\to F$ of complete locally convex
spaces is said to be nuclear if $f$ can be written as
$$f(x)=\sum\lambda_j\,\xi_j(x)\,y_j$$
where $(\lambda_j)$ is a sequence of scalars with $\sum|\lambda_j|<+\infty$,
$\xi_j\in E'$ an equi\-con\-tinuous sequence of linear forms and $y_j\in F$ a
bounded sequence.
\endclaim

When $E$ and $F$ are Banach spaces, the space of nuclear morphisms is
isomorphic to $E'\hotimes_\pi F$ and the nuclear norm $||f||_\nu$ is
defined to be the norm in this space, namely
$$||f||_\nu=\inf\Big\{\sum|\lambda_j|\,;~f=\sum\lambda_j\,\xi_j\otimes y_j,~
||\xi_j||\le 1,~||y_j||\le 1~\Big\}.\leqno(5.8)$$
For general spaces $E$, $F$, the equicontinuity of $(\xi_j)$ means that
there is a semi-norm $p$ on $E$ and a constant $C$ such that
$|\xi_j(x)|\le C\,p(x)$ for all $j$.  Then the definition shows that
$f:E\to F$ is nuclear if and only if $f$ can be factorized as $E\to
E_1\to F_1\to F$ where $E_1\to F_1$ is a nuclear morphism of Banach
spaces: indeed we need only take $E_1$ be equal to the Hausdorff
completion $\wh E_p$ of $(E,p)$ and let $F_1$ be the subspace of $F$
generated by the closed balanced convex hull of~$\{y_j\}$ ($=$ unit
ball in $F_1)\,$; moreover, if $u:S\to E$ and $v:F\to T$ are
continuous, the nuclearity of $f$ implies the nuclearity of $v\circ
f\circ u\,$; its nuclear decomposition is then $v\circ f\circ u=
\sum\lambda_j\,(\xi_j\circ u)\otimes v(y_j)$.

\claim{(5.9) Remark} \rm Every nuclear morphism is compact: indeed, we may assume
in Def.~5.7 that $(y_j)$ converges to $0$ and $\sum|\lambda_j|\le 1$,
otherwise we replace $y_j$ by $\varepsilon_j y_j$ with $\varepsilon_j$
converging to zero such that $\sum|\lambda_j/\varepsilon_j|\le 1\,$;
then, if $U\subset F$ is  a neighborhood of $0$ such that
$|\xi_j(U)|\le 1$ for all $j$, the image $f(U)$ is contained in the
closed convex hull of the compact set $\{y_j\}\cup\{0\}$, which is
compact.
\endclaim

\claim{(5.10) Proposition} If $E,F,G$ are Banach spaces and if
$f:E\to F$ is nuclear, there is a continuous morphism
$$f\hotimes\Id_G:E\hotimes_\varepsilon G\longrightarrow F\hotimes_\pi G$$
extending $f\otimes\Id_G$, such that
$||f\hotimes\Id_G||\le||f||_\nu$.
\endclaim

\proof{Proof} If $f=\sum\lambda_j\,\xi_j\otimes y_j$ as in (5.8), then
for any $t\in E\otimes G$ we have
$$(f\otimes\Id_G)(t)=\sum\lambda_j\big(\xi_j\otimes\Id_G(t)\big)
\otimes y_j$$
where $(\xi_j\otimes\Id_G)(t)\in G$ has norm
$$||(\xi_j\otimes\Id_G)(t)||=
\sup_{\eta\in G',\,||\eta||\le 1}\big|\eta\big(\xi_j\otimes\Id_G(t)\big)\big|
=\sup_\eta\big|\xi_j\otimes\eta(t)\big|\le ||t||_\varepsilon.$$
Therefore $||f\otimes\Id_G(t)||_\pi\le\sum|\lambda_j|\,||t||_\varepsilon$,
and the infimum over all decompositions of $f$ yields
$$||f\otimes\Id_G(t)||_\pi\le||f||_\nu||t||_\varepsilon.$$
Proposition 5.10 follows.\qed
\endproof

If $E$ is a Fr\'echet space and $(p_j)$ an increasing sequence of
semi-norms on $E$ defining the topology of $E$, we have
$$E=\lim_{{\displaystyle\longleftarrow}}\wh E_{p_j},$$
where $\wh E_{p_j}$ is the Hausdorff completion of $(E,p_j)$ and
$\wh E_{p_{j+1}}\to\wh E_{p_j}$ the canonical morphism. Here
$\wh E_{p_j}$ is a Banach space for the induced norm $\wh p_j$.

\claim{(5.11) Definition} A Fr\'echet space $E$ is said to be nuclear if the
topology of $E$ can be defined by an increasing sequence of semi-norms
$p_j$ such that each canonical morphism
$$\wh E_{p_{j+1}}\longrightarrow\wh E_{p_j}$$
of Banach spaces is nuclear.
\endclaim

If $E,F$ are arbitrary locally convex spaces, we always have a
continuous morphism $E\hotimes_\pi F\to E\hotimes_\varepsilon F$, because
$p\otimes_\varepsilon q\le p\otimes_\pi q$. If $E$, say, is nuclear,
this morphism yields in fact an isomorphism $E\hotimes_\varepsilon F
\simeq E\hotimes_\pi F\,$: indeed, by Prop.~5.10, we have
$p_j\hotimes_\pi q\le C_j\,p_{j+1}\hotimes_\varepsilon q$ where $C_j$
is the nuclear norm of $\wh E_{p_{j+1}}\to\wh E_{p_j}$. Hence, when
$E$ or $F$ is nuclear, we will identify $E\hotimes_\pi F$ and
$E\hotimes_\varepsilon F$ and omit $\varepsilon$ or $\pi$ in the
notation $E\hotimes F$.

\claim{(5.12) Example} \rm Let $D=\prod D(0,R_j)$ be a polydisk in $\bC^n$.
For any $t\in{}]0,1[$, we equip $\cO(D)$ with the semi-norm
$$p_t(f)=\sup_{tD}|f|.$$
The completion of $\big(\cO(D),p_t\big)$ is the Banach space $E_t$ of
holomorphic functions on $tD$ which are continuous up to the boundary.
We claim that for $t'<t<1$ the restriction map
$$\rho_{t,t'}:E_{t'}\longrightarrow E_t$$
is nuclear. In fact, for $f\in\cO(D)$, we have $f(z)=\sum a_\alpha
z^\alpha$ where $a_\alpha=a_\alpha(f)$ satisfies the Cauchy inequalities
$|a_\alpha(f)|\le p_{t'}(f)/(t'R)^{\alpha}$ for all $\alpha\in\bN^n$. 
The formula $f=\sum a_\alpha(f)\,e_\alpha$ with $e_\alpha(z)=z^\alpha$ 
shows that
$$||\rho_{t,t'}||_\nu\le\sum||a_\alpha||_{p_{t'}}||e_\alpha||_{p_t}\le
\sum(t'R)^{-\alpha}(tR)^\alpha=(1-t/t')^{-n}<+\infty.$$
We infer that $\cO(D)$ is a nuclear Fr\'echet space. It is also in a
natural way a fully nuclear Fr\'echet algebra (see Def.~5.39 below).\qed
\endclaim

\claim{(5.13) Proposition} Let $E$ be a nuclear space. A morphism
$f:E\to F$ is nuclear if and only if $f$ admits a factorization
$E\to M\to F$ through a Banach space $M$.
\endclaim

\proof{Proof} By definition, a nuclear map $f:E\to F$ always has a factorization
through a Banach space (even if $E$ is not nuclear). Conversely, if $E$ is
nuclear, any continuous linear map $E\to M$ into a Banach space $M$ is
continuous for some semi-norm $p_j$ on $E$, so this map has a factorization
$$E\to\wh E_{p_{j+1}}\to\wh E_{p_j}\to M$$
in which the second arrow is nuclear. Hence any map $E\to M\to F$ is
nuclear.\qed
\endproof

\claim{(5.14) Proposition} \smallskip
\item{\rm a)} If $E$, $F$ are nuclear spaces, then $E\hotimes F$ is nuclear.
\smallskip
\item{\rm b)} Any closed subspace or quotient space of a nuclear space
is nuclear.
\smallskip
\item{\rm c)} Any countable product of nuclear spaces is nuclear.
\smallskip
\item{\rm d)} Any countable inverse limit of nuclear spaces is nuclear.
\vskip0pt
\endclaim

\proof{Proof} a) If $f:E_1\to F_1$ and $g:E_2\to F_2$ are nuclear morphisms of
Banach spaces, it is easy to check that $f\hotimes_\pi g$ and
$f\hotimes_\varepsilon g$ are nuclear with
$||f\hotimes_? g||_\nu\le||f||_\nu||g||_\nu$ in both cases. Property a)
follows by applying this to the canonical morphisms
$\wh E_{p_{j+1}}\to\wh E_{p_j}$ and $\wh F_{q_{j+1}}\to\wh F_{q_j}$.
\medskip
\noindent{c)} Let $E_k$, $k\in\bN$, be nuclear spaces and $F=\prod E_k$.
If $(p^k_j)$ is an increasing family of semi-norms on $E_k$ as in
Def.~5.11, then the topology of $F$ is defined by the family of
semi-norms
$$q_j(x)=\max_{0\le k\le j}p^k_j(x_k),~~~~x=(x_k)\in F.$$
Then $\wh F_{q_j}=\bigoplus_{0\le k\le j}\wh E_{k,p^k_j}$ and
$$\big(\wh F_{q_{j+1}}\to\wh F_{q_j}\big)=\bigoplus_{0\le k\le j}
\big(\wh E_{k,p^k_{j+1}}\to\wh E_{k,p^k_j}\big)\oplus
\big(\wh E_{j+1,p^{j+1}_{j+1}}\to\{0\}\big)$$
is easily seen to be nuclear.
\medskip
\noindent{b)} If $F\subset E$ is closed, then $\wh F_{p_j}$ can be
identified to a closed subspace of~$\wh E_{p_j}$, the map $\wh F_{p_{j+1}}
\to\wh F_{p_j}$ is the restriction of $\wh E_{p_{j+1}}\to\wh E_{p_j}$
and we have \hbox{$\wh{E/F}_{p_j}\simeq\wh E_{p_j}/\wh F_{p_j}$.}
It is not true in general that the restriction or quotient of a nuclear
morphism is nuclear, but this is true for a binuclear \hbox{
$=(\hbox{nuclear}\circ\hbox{nuclear})$} morphism, as shown by Lemma~5.15~b)
below. Hence $\wh F_{p_{2j+2}}\to\wh E_{p_{2j}}$ and $\wh{E/F}_{p_{2j+2}}
\to\wh{E/F}_{p_{2j}}$ are nuclear, so $(p_{2j})$ is a fundamental family
of semi-norms on $F$ or $E/F$, as required in Def.~5.11.
\medskip
\noindent{d)} follows immediately from b) and c), since
$\smash{\displaystyle\lim_{\displaystyle\longleftarrow}}E_k$ is a closed 
subspace of $\prod E_k$.\qed
\endproof

\claim{(5.15) Lemma} Let $E$, $F$, $G$ be Banach spaces.\smallskip
\item{\rm a)} If $f:E\to F$ is nuclear, then $f$ can be factorized
through a Hilbert space $H$ as a morphism $E\to H\to F$.
\smallskip
\item{\rm b)} Let $g:F\to G$ be another nuclear morphism. If
$\Im(g\circ f)$ is contained in a closed subspace $T$ of $G$, then
$g\circ f:E\to T$ is nuclear. If $\ker(g\circ f)$ contains a closed
subspace $S$ of $E$, the induced map $(g\circ f)^\sim:E/S\to G$ is
nuclear.\smallskip
\endclaim

\proof{Proof} a) Write $f=\sum_{j\in I}\xi_j\otimes y_j\in E'\hotimes_\pi F$ with
$\sum||\xi_j||\,||y_j||<+\infty$. Without loss of generality, we may
suppose $||\xi_j||=||y_j||$. Then $f$ is the composition
$$E\longrightarrow\ell^2(I)\longrightarrow F,~~~~x\longmapsto\big(\xi_j(x)\big),~~
(\lambda_j)\longmapsto\sum\lambda_jy_j.$$
b) Decompose $g$ into $g=v\circ u$ as in a) and write $g\circ f$ 
as the composition
$$E\buildo f\over\longrightarrow F\buildo u\over\longrightarrow H\buildo v\over\longrightarrow G$$
where $H$ is a Hilbert space. If $\Im(g\circ f)\subset T$ and if
$T\subset G$ is closed, then $H_1=v^{-1}(T)$ is a closed subspace of
$H$ containing $\Im(u\circ f)$. Therefore $g\circ f:E\to T$ is the composition
$$E\buildo f\over\longrightarrow F\buildo u\over\longrightarrow H\buildo{\rm pr}^\perp\over\longrightarrow
H_1\buildo v_{\restriction H_1}\over\longrightarrow T$$
where $f$ is nuclear and $g\circ f:E\longrightarrow T$ is nuclear. Similar
proof for $(g\circ f)^\sim:E/S\to G$ by using decompositions 
$f=v\circ u:E\to H\to F$ and
$$(g\circ f)^\sim:E/S\buildo\wt u\over\longrightarrow H/H_1\simeq H_1^\perp
\buildo v_{\restriction H_1^\perp}\over\longrightarrow F\buildo g\over\longrightarrow G$$
where $H_1=\ol{u(S)}$ satisfies $H_1\subset\ker(g\circ v)\subset H$.\qed
\endproof

\claim{(5.16) Corollary} Let $E$ be a nuclear space and let $E\to F$ be a
nuclear morphism.
\smallskip
\item{\rm a)} If $f(E)$ is contained in a closed subspace $T$ of $F$, then
the morphism \hbox{$f_1:E\to T$} induced by $f$ is nuclear.
\smallskip
\item{\rm b)} If $\ker f$ contains a closed subspace $S$ of $E$, then
$\wt f:E/S\to F$ is nuclear.
\endclaim

\proof{Proof} Let $E\buildo u\over\longrightarrow M\buildo v\over\longrightarrow F$ be a factorization 
of $f$ through a Banach space~$M$. In case a), resp.\ b), $M_1=v^{-1}(T)$
is a closed subspace of $M$, resp.\ $M/\ol{u(S)}$ is a Banach space,
and we have factorizations
$$f_1:E\buildo u_1\over\longrightarrow M_1\buildo v_1\over\longrightarrow T,~~~~
\wt f:E/S\buildo\wt u\over\longrightarrow M/\ol{u(S)}\buildo\wt v\over\longrightarrow F$$
where $u_1$, $\wt u$ are induced by $u$ and $v_1$, $\wt v$ by $v$. Hence
$f_1$ and $\wt f$ are nuclear.\qed
\endproof

\claim{(5.17) Proposition} Let $0\to E_1\to E_2\to E_3\to 0$ be an
exact sequence of Fr\'echet spaces and let $F$ be a Fr\'echet space.
If $E_2$ or $F$ is nuclear, there is an exact sequence
$$0\longrightarrow E_1\hotimes F\longrightarrow E_2\hotimes F\longrightarrow E_3\hotimes F\longrightarrow 0.$$
\endclaim

\proof{Proof} If $E_2$ is nuclear, then so are $E_1$ and $E_3$ by Prop.~5.14~b).
Hence $E_1\hotimes F\to E_2\hotimes F$ is a monomorphism and 
$E_2\hotimes F\to E_3\hotimes F$ an epi\-mor\-phism by Prop.~5.6 and 5.5.
It only remains to show that
$$\Im\big(E_1\hotimes F\longrightarrow E_2\hotimes F\big)=
\ker\big(E_2\hotimes F\longrightarrow E_3\hotimes F\big)$$
and for this, we need only show that the left hand side is dense in the
right hand side (we already know it is closed). Let
$\varphi\in(E_2\hotimes F)'$ be a linear form, viewed as a continuous
bilinear form on $E_2\times F$. If $\varphi$ vanishes on the image
of $E_1\hotimes F$, then $\varphi$ induces a continuous bilinear form
on $E_3\times F$ by passing to the quotient. Hence $\varphi$ must
vanish on the kernel of $E_2\hotimes F\to E_3\hotimes F$, and
our density statement follows by the Hahn-Banach theorem.\qed
\endproof

\subsection{5.B}{K\"unneth Formula for Coherent Sheaves}
As an application of the above general concepts, we now show how 
topological tensor products can be used to compute holomorphic 
functions and cohomology of coherent sheaves on product spaces.

\claim{(5.18) Proposition} Let $\cF$ be a coherent analytic
sheaf on a complex analytic scheme $(X,\cO_X)$. Then $\cF(X)$ is
a nuclear space.
\endclaim

\proof{Proof} Let $A\subset\Omega\subset\bC^N$ be an open patch of $X$ such that
the image sheaf $(i_A)_\star\cF_{\restriction A}$ on $\Omega$ has a
resolution
$$\cO_\Omega^{p_1}\longrightarrow\cO_\Omega^{p_0}\longrightarrow(i_A)_\star\cF_{\restriction A}\longrightarrow0$$
and let $D\compact\Omega$ be a polydisk. As $D$ is Stein, we get an exact
sequence
$$\cO^{p_1}(D)\longrightarrow\cO^{p_0}(D)\longrightarrow\cF(A\cap D)\longrightarrow 0.\leqno(5.19)$$
Hence $\cF(A\cap D)$ is a quotient of the nuclear space $\cO^{p_0}(D)$
and so $\cF(A\cap D)$ is nuclear by (5.14~b). Let $(U_\alpha)$ be a 
countable covering of $X$ by open sets of the form $A\cap D$. Then $\cF(X)$ 
is a closed subspace of $\prod\cF(U_\alpha)$, thus $\cF(X)$ is nuclear
by \hbox{(5.14~b,c)}.\qed
\endproof

\claim{(5.20) Proposition} Let $\cF$, $\cG$ be coherent
sheaves on complex analytic schemes $X$, $Y$ respectively.
Then there is a canonical isomorphism
$$\cF\stimes\cG(X\times Y)\simeq\cF(X)\hotimes\cG(Y).$$
\endclaim

\proof{Proof} We show the proposition in several steps of increasing generality.
\medskip
\item{a)}{\it $X=D\subset\bC^n$, $Y=D'\subset\bC^p$ are
polydisks, $\cF=\cO_X$, $\cG=\cO_Y$.}\smallskip
Let $p_t(f)=\sup_{tD}|f|$, $p'_t(f)=\sup_{tD'}|f|$ and
$q_t(f)=\sup_{t(D\times D')}|f|$ be the semi-norms
defining the topology of $\cO(D)$, $\cO(D')$ and $\cO(D\times D')$, 
respectively.
Then $\wh E_{p_t}$ is a closed subspace of the space $C(t\ol D)$ of 
continuous functions on $t\ol D$ with the sup norm, and we have
$p_t\otimes_\varepsilon p'_t=q_t$ by example (5.3~c). Now, 
$\cO(D)\otimes\cO(D')$ is dense in $\cO(D\times D')$, hence its
completion with respect to the family $(q_t)$ is
$\cO(D)\hotimes_\varepsilon\cO(D')=\cO(D\times D')$.
\medskip
\item{b)}{\it $X$ is embedded in a polydisk $D\subset\bC^n$,
$X=A\cap D\buildo i\over\lhra D$,\Fbreak
$i_\star\cF$ is the cokernel of a morphism 
$\cO_D^{p_1}\longrightarrow\cO_D^{p_0}$,\Fbreak
$Y=D'\subset\bC^p$ is a polydisk and $\cG=\cO_Y$.}\smallskip
\noindent{}By taking the external tensor product with $\cO_Y$, we get an exact 
sequence
$$\cO_{D\times Y}^{p_1}\longrightarrow\cO_{D\times Y}^{p_0}\longrightarrow i_\star\cF\stimes
\cO_Y\longrightarrow 0.\leqno(5.21)$$
Then we find a commutative diagram
$$\cmalign{
\cO^{p_1}(D)&\hotimes\cO(Y)&~\longrightarrow~\cO^{p_0}(D)&\hotimes\cO(Y)&~\longrightarrow~
\hfill\cF(X)&\hotimes\cO(Y)&~\longrightarrow~ 0\cr
\noalign{\vskip4pt}
&~\big\downarrow\simeq&&~\big\downarrow\simeq&&~\big\downarrow&\cr
\noalign{\vskip4pt}
\hfill\cO^{p_1}(D&\times Y)&~\longrightarrow~\hfill\cO^{p_0}(D&\times Y)&~\longrightarrow~
\cF\stimes\cO_Y&(X\times Y)&~\longrightarrow~ 0\cr}$$
in which the first line is exact as the image of (5.19) by the exact functor
$\bu\hotimes\cO(Y)$, and the second line is exact because the exact sequence 
of sheaves (5.21) gives an exact sequence of spaces of sections on the Stein 
space $D\times Y\,$; note that $i_\star\cF\stimes\cO_Y(D\times Y)=
\cF\stimes\cO_Y(X\times Y)$. As the first two vertical arrows are isomorphisms
by a), the third one is also an isomorphism.
\medskip
\item{c)}{\it $X$, $\cF$ are as in {\rm b)},\Fbreak
$Y$ is embedded in a polydisk $D'\subset\bC^p$,
$\smash{Y=A'\cap D'\buildo j\over\lhra D'}$\Fbreak
and $j_\star\cG$ is the cokernel of $\cO_{D'}^{q_1}\longrightarrow\cO_{D'}^{q_0}$.}
\smallskip
\noindent{}Taking the external tensor product with $\cF$, we get an exact sequence
$$\cF\stimes\cO_{D'}^{q_1}\longrightarrow\cF\stimes\cO_{D'}^{q_0}\longrightarrow\cF\stimes 
j_\star\cG\longrightarrow 0$$
and with the same arguments as above we obtain a commutative diagram
$$\cmalign{
\hfill\cF(X)&\hotimes\cO^{q_1}(D')&~\longrightarrow~\hfill\cF(X)&\hotimes\cO^{q_0}(D')
&~\longrightarrow~\hfill\cF(X)&\hotimes\cG(Y)&~\longrightarrow~ 0\cr
\noalign{\vskip4pt}
&~\big\downarrow\simeq&&~\big\downarrow\simeq&&~\big\downarrow&\cr
\noalign{\vskip4pt}
\hfill\cF\stimes\cO^{q_1}_{D'}&(X\times D')&~\longrightarrow~\hfill\cF\stimes
\cO^{q_0}_{D'}&(X\times D')&~\longrightarrow~\cF\stimes\cG&(X\times Y)&~\longrightarrow~ 0.\cr}$$
\item{d)}{\it $X$, $\cF$ are as in {\rm b),c)} and $Y$, $\cG$ are
arbitrary.}\smallskip
Then $Y$ can be covered by open sets $U_\alpha=A_\alpha\cap D_\alpha$ 
embedded in polydisks $D_\alpha$, on which the image of $\cG$ admits a two-step
resolution. We have $\cF\stimes\cG(X\times U_\alpha)\simeq
\cF(X)\hotimes\cG(U_\alpha)$ by c), and the same is true over the
intersections $X\times U_{\alpha\beta}$ because $U_{\alpha\beta}=U_\alpha
\cap U_\beta$ can be embedded by the cross product embedding
$j_\alpha\times j_\beta:U_{\alpha\beta}\to D_\alpha\times D_\beta$.
We have an exact sequence
$$0\longrightarrow\cG(Y)\longrightarrow\prod_\alpha\cG(U_\alpha)\longrightarrow\prod_{\alpha,\beta}
\cG(U_{\alpha\beta})$$
where the last arrow is $(c_\alpha)\mapsto(c_\beta-c_\alpha)$, and a
commutative diagram with exact lines
$$\cmalign{
&0\longrightarrow\hfill\cF(X)&\hotimes\cG(Y)&\longrightarrow\hfill\prod\cF(X)&\hotimes\cG(U_\alpha)
&\longrightarrow\prod\hfill\cF(X)&\hotimes\cG(U_{\alpha\beta})\cr
&&~\big\downarrow&&~\big\downarrow\simeq&&~\big\downarrow\simeq\cr
&0\longrightarrow\cF\stimes\cG(&X\times Y)&\longrightarrow\prod\cF\stimes\cG(&X\times U_\alpha)
&\longrightarrow\prod\cF\stimes\cG(&X\times U_{\alpha\beta}).\cr}$$
Therefore the first vertical arrow is an isomorphism.
\medskip
\item{e)}{\it $X$, $\cF$, $Y$, $\cG$ are arbitrary.}\smallskip
This case is treated exactly in the same way as d) by reversing the 
roles of $\cF$, $\cG$ and by using d) to get the isomorphism in the last
two vertical arrows.\qed
\endproof

\claim{(5.22) Corollary} Let $\cF$, $\cG$ be coherent sheaves over complex
analytic schemes $X$, $Y$ and let $\pi:X\times Y\to X$ be the projection.
Suppose that $H^\bu(Y,\cG)$ is Hausdorff.
\smallskip
\item{\rm a)} If $X$ is Stein, then
$H^q(X\times Y,\cF\stimes\cG)\simeq\cF(X)\hotimes H^q(Y,\cG)$.
\smallskip
\item{\rm b)} In general, for every open set $U\subset X$,
$$\big(R^q\pi_\star(\cF\stimes\cG)\big)(U)=\cF(U)\hotimes H^q(Y,\cG).$$
\smallskip
\item{\rm c)} If $H^q(Y,\cG)$ is finite dimensional, then
$$R^q\pi_\star(\cF\stimes\cG)=\cF\otimes H^q(Y,\cG).$$
\endclaim

\proof{Proof} a) Let $\cV=(V_\alpha)$ be a countable Stein covering of $Y$. By 
the Leray theorem, $H^\bu(Y,\cG)$ is equal to the cohomology of the \v Cech 
complex $C^\bu(\cV,\cG)$. Similarly $X\times\cV=(X\times V_\alpha)$ is a
Stein covering of $X\times Y$ and we have
$$H^q(X\times Y,\cF\stimes\cG)=H^q\big(C^\bu(X\times\cV,\cF\stimes\cG)\big).$$
However, Prop.~5.20 shows that $C^\bu(X\times\cV,\cF\stimes\cG)=
\cF(X)\hotimes C^\bu(\cV,\cG)$. Our assumption that 
$C^\bu(\cV,\cG)$ has Hausdorff cohomology implies that the cocycle and
coboundary groups are (nuclear) Fr\'echet spaces, and that each cohomology 
group can be computed by means of short exact sequences in this category. By 
Prop.~5.17, we thus get the desired equality
$$H^q\big(C^\bu(X\times\cV,\cF\stimes\cG)\big)=
\cF(X)\hotimes H^q\big(C^\bu(\cV,\cG)\big).$$
\noindent{b)} The presheaf $U\mapsto\cF(U)\hotimes H^q(Y,\cG)$ is in fact
a sheaf, because the tensor product with the nuclear space $H^q(Y,\cG)$ 
preserves the exactness of all sequences
$$0\longrightarrow\cF(U)\longrightarrow\prod\cF(U_\alpha)\longrightarrow\prod\cF(U_{\alpha\beta})$$
associated to arbitrary coverings $(U_\alpha)$ of $U$. Property b)
thus follows from a) and from the fact that $R^q\pi_\star(\cF\stimes\cG)$ is
the sheaf associated to the presheaf $U\mapsto H^q(U\times Y,\cF\stimes\cG)$.
\medskip
\noindent{c)} is an immediate consequence of b), since the finite
dimensionality of $H^q(Y,\cG)$ implies that this space is Hausdorff.\qed
\endproof

\claim{(5.23) K\"unneth formula} Let $\cF$, $\cG$ be coherent 
sheaves over complex analytic schemes $X$, $Y$ and suppose that the 
cohomology spaces $H^\bu(X,\cF)$ and $H^\bu(Y,\cG)$ are Hausdorff. 
Then there is an isomorphism
$$\eqalign{
\bigoplus_{p+q=k}H^p(X,\cF)\hotimes H^q(Y,\cG)&\buildo\simeq\over\longrightarrow
H^k(X\times Y,\cF\stimes\cG)\cr
\bigoplus\alpha_p\otimes\beta_q&\buildo~\over\longmapsto
\sum\alpha_p\smallsmile\beta_q.\cr}$$
\endclaim

\proof{Proof} Consider the Leray spectral sequence associated to the coherent
sheaf $\cS=\cF\stimes\cG$ and to the projection $\pi:X\times Y\to X$. By 
Cor.~5.22~b) and a use of \v Cech cohomology, we find 
$$E_2^{p,q}=H^p(X,R^q\pi_\star\cF\stimes\cG)=H^p(X,\cF)\hotimes H^q(Y,\cG).$$
It remains to show that the Leray spectral sequence degenerates in
$E_2$. For this, we argue as in the proof of Th.~IV-15.9. In that proof,
we defined a morphism of the double complex $C^{p,q}=\cF^{[p]}(X)\otimes
\cG^{[q]}(Y)$ into the double complex that defines the Leray
spectral sequence (in IV-15.9, we only considered the sheaf theoretic
external tensor product $\cF\stimes\cG$, but there is an obvious morphism
of that one into the analytic tensor product). We get a morphism of
spectral sequences which induces at the $E_2$-level the obvious morphism
$$H^p(X,\cF)\otimes H^q(Y,\cG)\longrightarrow H^p(X,\cF)\hotimes H^q(Y,\cG).$$
It follows that the Leray spectral sequence $E^{p,q}_r$ is
obtained for $r\ge 2$ by taking the completion of the spectral
sequence of $C^{\bu,\bu}$. Since this spectral sequence degenerates
in $E_2$ by the algebraic K\"unneth theorem, the Leray spectral
sequence also satisfies $d_r=0$ for $r\ge 2$.\qed
\endproof

\claim{(5.24) Remark} \rm If $X$ or $Y$ is compact, the K\"unneth formula holds
with $\otimes$ instead of $\hotimes$, and the assumption that both
cohomology spaces are Hausdorff is unnecessary. The proof is exactly
the same, except that we use (5.22~c) instead of (5.22~b).
\endclaim

\subsection{5.C}{Modules over Nuclear Fr\'echet Algebras}
Throughout this subsection, we work in the category of nuclear
Fr\'echet spaces. Recall that a topological algebra (commutative, with
unit element~$1$) is an algebra $A$ together with a topological vector
space structure such that the multiplication $A\times A\to A$ is
continuous. $A$ is said to be a Fr\'echet (resp.\ nuclear) algebra if
it is Fr\'echet (resp.\ nuclear) as a topological vector space.

\claim{(5.25) Definition} A $($Fr\'echet, resp.\ nuclear$)$ $A$-module $E$ is a
$($Fr\'echet, resp.\ nuclear$)$ space $E$ with a $A$-module structure
such that the multiplication $A\times E\to E$ is continuous. The
module $E$ is said to be nuclearly free if $E$ is of the form $A\hotimes V$
where $V$ is a nuclear Fr\'echet space.
\endclaim

Assume that $A$ is nuclear and let $E$ be a nuclear $A$-module.
A {\it nuclearly free resolution} $L_\bu$ of $E$ is an exact sequence
of $A$-modules and continuous $A$-linear morphisms
$$\cdots\longrightarrow L_q\buildo d_q\over\longrightarrow L_{q-1}\longrightarrow\cdots\longrightarrow L_0\longrightarrow E\longrightarrow 0
\leqno(5.26)$$
in which each $L_q$ is a nuclearly free $A$-module. Such a resolution is
said to be {\it direct} if each map $d_q$ is direct, i.e.\ if $\Im d_q$
has a topological supplementary space in $L_{q-1}$ (as a vector space 
over $\bR$ or $\bC$, not necessarily as a $A$-module).

\claim{(5.27) Proposition} Every nuclear $A$-module $E$ admits a 
direct nuclearly free resolution.
\endclaim

\proof{Proof} We define the ``standard resolution" of $E$ to be
$$L_q=A\hotimes\ldots\hotimes A\hotimes E$$ 
where $A$ is repeated \hbox{$(q+1)$} times; the $A$-module structure of 
$L_q$ is chosen to be the one given by the first factor and we set 
$d_0(a_0\otimes x)=a_0x$,
$$\eqalign{
d_q(a_0\otimes\ldots\otimes a_q\otimes x)=
\sum_{0\le i<q}&(-1)^ia_0\otimes\ldots\otimes a_ia_{i+1}\otimes\ldots
\otimes a_q\otimes x\cr
{}+{}&(-1)^qa_0\otimes\ldots\otimes a_{q-1}\otimes a_qx.\cr}$$
Then there is a homotopy operator $h_q:L_q\to L_{q+1}$ given by 
$h_q(t)=1\otimes t$ for all $q$ ($h_q$, however, is not $A$-linear). 
This implies easily that $L_\bu$ is a direct nuclearly free resolution.\qed
\endproof

If $E$ and $F$ are two nuclear $A$-modules, we define $E\hotimes_A F$
to be
$$\leqalignno{
&E\hotimes_A F=\hbox{\rm coker}\big(E\hotimes A\hotimes F\buildo d\over\longrightarrow
E\hotimes F\big)~~~~\hbox{\rm where}&(5.28)\cr
&d(x\otimes a\otimes y)=ax\otimes y-x\otimes ay.\cr}$$
Then $E\hotimes_A F$ is a $A$-module which it is not necessarily
Hausdorff. If \hbox{$E\hotimes_A F$} is Hausdorff, it is in fact
a nuclear $A$-module by Prop.~5.14. If $E$ is nuclearly free, say 
$E=A\hotimes V\simeq V\hotimes A$, we have $E\hotimes_A F=V\hotimes F$ 
(which is thus Hausdorff): indeed, there is an exact sequence
$$\eqalign{
&V\hotimes A\hotimes A\hotimes F\longrightarrow V\hotimes A\hotimes F\longrightarrow V\hotimes F
\longrightarrow 0,\cr
&v\otimes a_0\otimes a_1\otimes x\longmapsto v\otimes a_0a_1\otimes x-
v\otimes a_0\otimes a_1x,~~~~v\otimes a\otimes x\longmapsto v\otimes ax,\cr}$$
obtained by tensoring the standard resolution of $F$ with $V\hotimes\,$;
observe that the tensor product $\hotimes$ with a nuclear space preserves
exact sequences thanks to Prop.~5.17. We further define $\toor_q^A(E,F)$ to be
$$\toor_q^A(E,F)=H_q(E\hotimes_A L_\bu),\leqno(5.29)$$
where $L_\bu$ is the standard resolution of $F$. There is in fact an 
isomorphism
$$\eqalign{
E\hotimes_A L_\bu&\buildo\simeq\over\longrightarrow 
E\hotimes A\hotimes\cdots\hotimes A\hotimes F\cr
x\otimes_A(a_0\otimes a_1\otimes\ldots\otimes a_q\otimes y)
&\buildo\over\longmapsto a_0x\otimes a_1\otimes\ldots\otimes a_q\otimes y\cr}$$
where $A$ is repeated $q$ times in the target space. In this
isomorphism, the differential becomes
$$\eqalign{
&d_q(x\otimes a_1\otimes\ldots\otimes a_q\otimes y)=
a_1x\otimes a_2\otimes\ldots\otimes a_q\otimes y\cr
&\qquad\qquad\qquad{}+\sum_{1\le i<q}(-1)^i
x\otimes a_1\otimes\ldots\otimes a_ia_{i+1}\otimes\ldots\otimes a_q\otimes y\cr
&\qquad\qquad\qquad{}+(-1)^qx\otimes a_1\otimes\ldots\otimes a_{q-1}
\otimes a_qy.\cr}$$
In particular, we get $\toor_0^A(E,F)=E\hotimes_A F$. Moreover, if we
exchange the roles of $E$ and $F$, we obtain a complex which is
isomorphic to the above one up to the sign of $d_q$, hence
$\toor_q^A(E,F)\simeq\toor_q^A(F,E)$. If $E=A\hotimes V$ is nuclearly
free, the complex $E\hotimes_A L_\bu=V\hotimes L_\bu$ is exact, thus
$$\hbox{\rm$E$ or $F$ nuclearly free}\Longrightarrow
\toor_q^A(E,F)=0~~\hbox{\rm for $q\ge 1$.}$$

\claim{(5.30) Proposition} For any exact sequence
$0\to E_1\to E_2\to E_3\to 0$ of nuclear $A$-modules and any nuclear
$A$-module $F$, there is an $($algebraic$)$ exact sequence
$$\cmalign{
\hfill\cdots\toor_q^A(E_1,F)&\longrightarrow\toor_q^A(E_2,F)&\longrightarrow\toor_q^A(E_3,F)&\longrightarrow
\toor_{q-1}^A(E_1,F)\cdots\cr
\longrightarrow~~E_1\hotimes_A F~~&\longrightarrow~~E_2\hotimes_A F&\longrightarrow~~E_3\hotimes_A F
&\longrightarrow 0.\cr}$$
\endclaim

\proof{Proof} As the standard resolution $L_\bu\to F$ is nuclearly free,
$L_q=A\hotimes V_q$ say, then $E_j\hotimes_A L_\bu=E_j\hotimes V_\bu$
for $j=1,2,3$, so we have a short exact sequence of complexes
$$0\longrightarrow E_1\hotimes_A L_\bu\longrightarrow E_2\hotimes_A L_\bu\longrightarrow E_3\hotimes_A
L_\bu\longrightarrow 0.\eqno{\square}$$
\endproof

\claim{(5.31) Corollary} For any nuclearly free $($possibly non direct$)$
resolution $L_\bu$ of $F$, there is a canonical isomorphism
$$\toor_q^A(E,F)\simeq H_q(E\hotimes_A L_\bu).$$
\endclaim

\proof{Proof} Set $B_q=\Im(L_{q+1}\to L_q)$ for all $q\ge 0$ and $B_{-1}=F$. 
Then apply (5.30) to the short exact sequences $0\to B_q\to L_q\to
B_{q-1}\to 0$ and the fact that $L_q$ is nuclearly free to get
$$\toor_k^A(E,B_{q-1})\simeq\cases{
\toor_{k-1}^A(E,B_q)&for $k>1$,\cr
\ker(E\hotimes_A B_q\to E\hotimes_A L_q)&for $k=1$.\cr}$$
Hence we obtain inductively
$$\eqalign{
\toor_q^A(E,F)&=\toor_q^A(E,B_{-1})\simeq\ldots\simeq\toor_1^A(E,B_{q-2})\cr
&\simeq\ker(E\hotimes_A B_{q-1}\to E\hotimes_A L_{q-1})\cr}$$
and a commutative diagram
$$\eqalign{
&E\hotimes_A L_{q+1}\longrightarrow E\hotimes_A L_q\longrightarrow E\hotimes_A B_{q-1}\longrightarrow 0\cr
&\qquad\searrow\qquad\qquad\nearrow\cr
&~~~~\qquad E\hotimes_A B_q\cr}$$
in which the horizontal line is exact (thanks to the surjectivity of
the left oblique arrow and the exactness of the sequence with
$E\hotimes_A B_q$ as first term). Therefore
$\ker(E\hotimes_A B_{q-1}\to E\hotimes_A L_{q-1})$
can be interpreted as the kernel of $E\hotimes_A L_q\to E\hotimes_A L_{q-1}$
modulo the image of $E\hotimes_A L_{q+1}\to E\hotimes_A L_q$,
and this is is precisely the definition of $H_q(E\hotimes_A L_\bu)$.\qed
\endproof

Now, we are ready to introduce the crucial concept of transversality.

\claim{(5.32) Definition} We say that two nuclear $A$-modules $E$, $F$ are
transverse if $E\hotimes_A F$ is Hausdorff and if $\toor_q^A(E,F)=0$
for $q\ge 1$.
\endclaim

For example, a nuclearly free $A$-module $E=A\hotimes V$ is transverse to
any nuclear $A$-module~$F$. Before proving further general properties, we 
give a fundamental example.

\claim{(5.33) Proposition} Let $X$, $Y$ be Stein spaces and let
$U'\subset U\compact X$, $V\compact Y$ be Stein open subsets. If $\cF$ is
a coherent sheaf over $X\times Y$, then $\cO(U')$ and $\cF(U\times V)$ are
transverse over $\cO(U)$. Moreover
$$\cO(U')\hotimes_{\cO(U)}\cF(U\times V)=\cF(U'\times V).$$
\endclaim

\proof{Proof} Let $\cL_\bu\to\cF$ be a free resolution of $\cF$ over $U\times V\,$;
such a resolution exists by Cartan's theorem~A. Then $\cL_\bu(U\times V)$
is a resolution of \hbox{$\cF(U\times V)$} which is nuclearly free over
$\cO(U)$, for $\cO(U\times V)=\cO(U)\hotimes\cO(V)\,$; in particular,
we get
$$\eqalign{
\cO(U')\hotimes_{\cO(U)}\cO(U\times V)&=\cO(U')\hotimes\cO(V)=\cO(U'\times V),\cr
\cO(U')\hotimes_{\cO(U)}\cL_\bu(U\times V)&=\cL_\bu(U'\times V).\cr}$$
But $\cL_\bu(U'\times V)$ is a resolution of $\cF(U'\times V)$, so
$$\toor_q^{\cO(U)}\big(\cO(U'),\cF(U\times V)\big)=\cases{
\cF(U'\times V)&for $q=0$,\cr
              0&for $q\ge 1$.\cr}\eqno{\square}$$
\endproof

\claim{(5.34) Properties} \smallskip
\item{\rm a)} If $0\to E_1\to E_2\to E_3\to 0$ is an exact sequence of
nuclear $A$-modules and if $E_2$, $E_3$ are transverse to $F$, then $E_1$
is transverse to $F$.
\smallskip
\item{\rm b)} Let $A\to A_1\to A_2$ be homomorphisms of nuclear algebras and 
let $E$ be a nuclear $A$-module. if $A_1$ and $A_2$ are transverse
to $E$ over $A$, then $A_2$ is tranverse to $A_1\hotimes_A E$ over $A_1$.
\smallskip
\item{\rm c)} Let $E^\bu$ be a complex of nuclear $A$-modules, bounded on
the right side, and let $M$ be a nuclear $A$-module which is transverse to
all $E^n$. If $E^\bu$ is acyclic in degrees $\ge k$, then $M\hotimes_A E^\bu$
is also acyclic in degrees $\ge k$.
\smallskip
\item{\rm d)} Let $E^\bu$, $F^\bu$ be complexes of nuclear $A$-modules, 
bounded on the right side. Let $f^\bu:E^\bu\to F^\bu$ be a $A$-linear
morphism and let $M$ be a nuclear $A$-module which is transverse to
all $E^q$ and $F^q$. If $f^\bu$ induces an isomorphism
$H^q(f^\bu):H^q(E^\bu)\to H^q(F^\bu)$ in degrees $q\ge k$ and an
epimorphism in degree $q=k-1$, then 
$$\Id_M\hotimes_A f^\bu:M\hotimes_A E^\bu\to M\hotimes_A F^\bu$$
has the same property.\smallskip
\endclaim

\proof{Proof} a) is an immediate consequence of the $\toor$ exact sequence.
\medskip
\noindent{}To prove b), we need only check that if $A_1$ is transverse to
$E$ over $A$, then
$$\toor_q^{A_1}(A_2,A_1\hotimes_A E)=\toor_q^A(A_2,E),~~~~\forall n\ge 0.$$
Indeed, if $L_\bu=A\hotimes V_\bu$ is a nuclearly free resolution of $E$
over $A$, then $A_1\hotimes_A L_\bu=A_1\hotimes V_\bu$ is a nuclearly free
resolution of $A_1\hotimes_A E$ over $A_1$, since $H_q(A_1\hotimes_A L_\bu)=
\toor_q^A(A_1,E)=0$ for $q\ge 1$. Hence
$$\eqalign{
\toor_q^{A_1}(A_2,A_1\hotimes_A{}\!E)&=H_q\big(A_2\hotimes_{A_1}{}\!(A_1
\hotimes_A L_\bu)\big)=H_q\big(A_2\hotimes_{A_1}{}\!(A_1\hotimes V_\bu)\big)\cr
&=H_q(A_2\hotimes V_\bu)=H_q(A_2\hotimes_A L_\bu)=\toor_q^A(A_2,E).\cr}$$
c) The short exact sequences $0\to Z^q(E^\bu)\lhra E^q\buildo d^q\over\longrightarrow 
Z^{q+1}(E^\bu)\to 0$ show by backward induction on $q$ that
$M$ is transverse to $Z^q(E^\bu)$ for \hbox{$q\ge k-1$}. Hence for
$q\ge k-1$ we obtain an exact sequence
$$0\longrightarrow M\hotimes_A Z^q(E^\bu)\lhra M\hotimes_A E^q\buildo d^q\over\longrightarrow 
M\hotimes_A Z^{q+1}(E^\bu)\longrightarrow 0,$$
which gives in particular $Z^q(M\hotimes_A E^\bu)=B^q(M\hotimes_A E^\bu)=
M\hotimes_A Z^q(E^\bu)$ for $q\ge k$, as desired.
\medskip
\noindent{d)} is obtained by applying c) to the {\it mapping cylinder}
$C(f^\bu)$, as defined in the following lemma (the proof is 
straightforward and left to the reader).\qed
\endproof

\claim{(5.35) Lemma} If $f^\bu:E^\bu\to F^\bu$ is a morphism of complexes,
the mapping cylinder $C^\bu=C(f^\bu)$ is the complex defined by
$C^q=E^q\oplus F^{q-1}$ with differential 
$$\pmatrix{d^q_E&0\cr -f^q&d^{q-1}_F\cr}:E^q\oplus F^{q-1}\longrightarrow
E^{q+1}\oplus F^q.$$
Then there is a short exact sequence $0\to F^{\bu-1}\to C^\bu\to E^\bu\to 0$
and the associated connecting homomorphism $\partial^q:H^q(E^\bu)\to
H^q(F^\bu)$ is equal to $H^q(f^\bu)\,$; in particular,
$C^\bu$ is acyclic in degree $q$ if and only if $H^q(f^\bu)$ is
injective and $H^{q-1}(f^\bu)$ is surjective.\qed
\endclaim

\subsection{5.D}{$A$-Subnuclear Morphisms and Perturbations}
We now introduce a notion of nuclearity relatively to an algebra $A$. This
notion is needed for example to describe the properties of the $\cO(S)$-linear
restriction map $\cO(S\times U)\to\cO(S\times U')$ when $U'\compact U$.

\claim{(5.36) Definition} Let $E$ and $F$ be Fr\'echet $A$-modules over a
Fr\'echet algebra $A$ and let $f:E\to F$ be a $A$-linear map. We say that
\smallskip
\item{\rm a)} $f$ is $A$-nuclear if there exist a scalar sequence
$(\lambda_j)$ with $\sum|\lambda_j|<+\infty$, an equicontinuous family
of $A$-linears maps $\xi_j:E\to A$ and a bounded sequence $y_j$ in $F$
such that for all $x\in E$
$$f(x)=\sum\lambda_j\,\xi_j(x)y_j.$$
\item{\rm b)} $f$ is $A$-subnuclear if there exists a Fr\'echet $A$-module
$M$ and an epimorphism $p:M\to E$ such that $f\circ p$ is $A$-nuclear; if
$E$ is nuclear, we also require $M$ to be nuclear.
\smallskip
\endclaim

If $f:E\to F$ is $A$-nuclear and if $u:S\to E$ and $v:F\to T$ are continuous
$A$-linear maps then $v\circ f\circ u$ is $A$-nuclear; the same is true for
$A$-subnuclear maps. If $V$ and $W$ are nuclear spaces and if 
$u:V\to W$ is $\bC$-nuclear, then $\Id_A\hotimes u:A\hotimes V\to 
A\hotimes W$ is $A$-nuclear. From this we infer:

\claim{(5.37) Proposition} Let $S$, $Z$ be Stein spaces and let
$U'\compact U\compact Z$ be Stein open subsets. Then the
restriction $\rho:\cO(S\times U)\to\cO(S\times U')$ is
\hbox{$\cO(S)$-nuclear}. If $\cF$ is a coherent sheaf over $Y\times Z$ with
$Y$ Stein and $S\compact Y$, then the restriction map
$\rho:\cF(S\times U)\to\cF(S\times U')$ is $\cO(S)$-subnuclear.
\endclaim

\proof{Proof} As $\cO(S\times U)=\cO(S)\hotimes\cO(U)$ and $\cO(U)\to\cO(U')$
is $\bC$-nuclear, only the second statement needs a proof. By Cartan's
theorem A, there exists a free resolution $\cL_\bu\to \cF$ over $S\times U$.
Then there is a commutative diagram
$$\cmalign{
\hfill\cL_0(S&\times U)&\longrightarrow\cF(S&\times U)\cr
\hfill\rho&\big\downarrow&&\big\downarrow\rho\cr
\hfill\cL_0(S&\times U')&\longrightarrow\cF(S&\times U')\cr}$$
in which the top horizontal arrow is an $\cO(S)$-epimorphism and the left
vertical arrow is an $\cO(S)$-nuclear map; its composition with the
bottom horizontal arrow is thus also $\cO(S)$-nuclear.\qed
\endproof

Let $f:E\to F$ be a $A$-linear morphism of Fr\'echet $A$-modules. Suppose
that $f(E)\subset F_1$ where $F_1$ is a {\it closed} $A$-submodule of $F$
and let $f_1:E\to F_1$ be the map induced by $f$. If $f$ is $A$-nuclear,
it is not true in general that $f_1$ is $A$-nuclear or $A$-subnuclear,
even if $A$, $E$, $F$ are nuclear. However:

\claim{(5.38) Proposition} With the above notations, suppose
$A$, $E$, $F$ nuclear. Let $B$ be a nuclear Fr\'echet algebra and let
$\rho:A\to B$ be a $\bC$-nuclear \hbox{homomorphism}. Suppose that $B$ is
transverse to $E$, $F$ and $F/F_1$ over $A$. If~$f:E\to F$ is
$A$-subnuclear, then $\Id_B\hotimes_A f_1:B\hotimes_A E\to B
\hotimes_A F_1$ is $B$-subnuclear.
\endclaim

\proof{Proof} We first show that $\rho\hotimes_A f_1:E=A\hotimes_A E\to
B\hotimes_A F_1$ is $\bC$-nuclear. Since a quotient of a $\bC$-nuclear
map is $\bC$-nuclear by Cor.~5.16~b), we may suppose for this that
$f$ is $A$-nuclear. Write
$$\eqalign{
f(x)&=\sum\lambda_j\,\xi_j(x)y_j,~~~~\xi_j:E\to A,~~\sum|\lambda_j|<+\infty,
~~y_j\in F,\cr
\rho(t)&=\sum\mu_k\,\eta_k(t)b_k,~~~~\eta_k:A\to\bC,~~\sum|\mu_k|<+\infty,
~~b_k\in B\cr}$$
as in the definition of ($A$-)nuclearity. Then $\rho\hotimes_A f:E\longrightarrow
B\hotimes_A F$ is \hbox{$\bC$-nuclear}: for any $x\in E$, we have
$\rho(\xi_j(x))=\xi_j(x)\rho(1)$ in the $A$-module structure of $B$,
hence
$$\eqalign{
\rho\hotimes_A f(x)=\rho\otimes f(1\otimes x)
&=\sum\lambda_j\,\rho(\xi_j(x))\hotimes_A y_j\cr
&=\sum\lambda_j\mu_k\,(\eta_k\circ\xi_j)(x)\,b_k\hotimes_A y_j.\cr}$$
By our transversality assumptions, $B\hotimes_A F_1$ is a closed subspace
of $B\hotimes_A F$. As $\Im(\rho\hotimes_A f)\subset B\hotimes_A F_1$,
the induced map $\rho\hotimes_A f_1:E\to B\hotimes_A F_1$ is $\bC$-nuclear
by Cor.~5.16~a). Finally, there is a commutative diagram
$$\cmalign{
B&\hotimes E&\buildo\Id_B\hotimes(\rho\hotimes_A f_1)\over{\larex 60 }
B\hotimes(&B\hotimes_A F_1)\cr
&~\big\downarrow&&~\big\downarrow\cr
B&\hotimes_A E&\buildo\Id_B\hotimes_A f_1\over{\larex 60 }
\hfill B&\hotimes_A F_1\cr}$$
in which the vertical arrows are $B$-linear epimorphisms. The top horizontal
arrow is $B$-nuclear by the $\bC$-nuclearity of $\rho\hotimes_A f_1$,
hence $\Id_B\hotimes_A f_1$ is \hbox{$B$-subnuclear}.\qed
\endproof

Example~5.12  suggests the following definition (which is somewhat less
general than some other in current use, but sufficient for our purposes).

\claim{(5.39) Definition} We say that a Fr\'echet algebra $A$ is fully nuclear
if the topology of $A$ is defined by an increasing family
$(p_t)_{t\in{}]0,1[}$ of multiplicative semi-norms $\big($that is,
$p_t(xy)\le p_t(x)\,p_t(y)\,\big)$, such that the Banach algebra homomorphism 
$\wh A_{p_{t'}}\to\wh A_{p_t}$ is nuclear for all $t<t'<1$.
\endclaim

If $A$ is fully nuclear and $t\in{}]0,1]$, we define $A_t$ to be the
completion of $A$ equipped with the family of semi-norms $p_{\lambda t}$,
$\lambda\in{}]0,1[$. Then $A_t$ is again a fully nuclear algebra, and for
all $t<t'<1$ the canonical map $A_{t'}\to A_t$ is nuclear: indeed, for
$t\le u<u'<t'$, there is a factorization
$$A_{t'}\longrightarrow\wh A_{p_{u'}}\longrightarrow\wh A_{p_u}\longrightarrow A_t.$$
If $E$ is a nuclear $A$-module, we say that $E$ is fully $A$-transverse
if $E$ is transverse to all $A_t$ over $A$. Then by 5.34~b), each
nuclear space
$$E_t=A_t\hotimes_A E\leqno(5.40)$$
is a fully $A_t$-transverse $A_t$-module. If $f:E\to F$ is a morphism
of fully $A$-transverse nuclear modules, there is an induced map
$$f_t=\Id_{A_t}\hotimes_A f:E_t\longrightarrow F_t,~~~~\forall t\in{}]0,1].
\leqno(5.40')$$

\claim{(5.41) Example} \rm Let $X$ be a closed analytic subscheme of an open set
$\Omega\subset\bC^N$, $D=D(a,R)\compact\Omega$ a polydisk and
$U=D\cap X$. We have an epimorphism $\cO(D)\to\cO(U)$. Denote by
$\wt p_t$ the quotient semi-norm of $p_t(f)=\sup_{D(a,tR)}|f|$ on $\cO(U)$.
Then $\cO(U)$ equipped with $(\wt p_t)_{t\in{}]0,1[}$ is a fully nuclear
algebra, and $\cO(U)_t=\cO\big(D(a,tR)\cap X\big)$.

Now, let $Y$ be a Stein space, $V\compact Y$ a Stein open subset and $\cF$ a
coherent sheaf over $X\times Y$. Then Prop.~5.33 shows that
$\cF(U\times V)$ is a fully transverse nuclear $\cO(U)$-module.
\endclaim

\claim{(5.42) Subnuclear perturbation theorem} Let $A$ be a fully
nuclear algebra,
let $E$ and $F$ be two fully $A$-transverse nuclear $A$-modules and let
$f,u:E\to F$ be $A$-linear maps. Suppose that $u$ is $A$-subnuclear and
that $f$ is an epimorphism. Then for every $t<1$, the cokernel of
$$f_t-u_t:E_t\longrightarrow F_t$$
is a finitely generated $A_t$-module $($as an algebraic module; we do not
assert that the cokernel is Hausdorff$)$.
\endclaim

\proof{Proof} We argue in several steps. The first step is the following special
case.
\endproof

\claim{(5.43) Lemma} Let $B$ be a Banach algebra, $S$ a Fr\'echet $B$-module and
$v:S\to S$ a $B$-nuclear morphism. Then $\Coker(\Id_S-v)$ is a finitely
generated $B$-module.
\endclaim

\proof{Proof} Let $v(x)=\sum\lambda_j\,\xi_j(x)y_j$ be a $B$-nuclear
decomposition of~$v$. We have a factorization
$$v=\beta\circ\alpha:S\buildo\alpha\over\longrightarrow\ell^1(B)\buildo\beta\over\longrightarrow S$$
where $\alpha(x)=\big(\lambda_j\xi_j(x)\big)$ and $\beta(t_j)=\sum t_jy_j$.
Set $w=\alpha\circ\beta:\ell^1(B)\to\ell^1(B)$. As $\alpha$ is $B$-nuclear,
so is $w$, and $\alpha$, $\beta$ induce isomorphisms
$$\Coker(\Id_S-v)~{\raise-4pt\hbox{
$\scriptstyle\wt\alpha\atop\displaystyle\relbar\joinrel\longrightarrow$}
\atop\raise4pt\hbox{
$\displaystyle\longleftarrow\joinrel\relbar\atop\scriptstyle\wt\beta$}}
~\Coker\big(\Id_{\ell^1(B)}-w\big).$$
We are thus reduced to the case when $S$ is a Banach module. Then we write
$v=v'+v''$ with
$$v'(x)=\sum_{1\le j\le N}\lambda_j\,\xi_j(x)y_j,~~~~
v''(x)=\sum_{j>N}\lambda_j\,\xi_j(x)y_j.$$
For $N$ large enough, we have $||v''||<1$, hence $\Id_S-v''$ is an
automorphism and\break $\Coker(\Id_S-v'-v'')$ is generated by the classes
of $y_1,\ldots,y_N$.\qed
\endproof

\proof{Proof of Theorem 5.42} a) We may suppose that $E$ is
nuclearly free and that $u$ is $A$-nuclear, otherwise we replace
$f$, $u$ by their composition with
\hbox{$A\hotimes M\longrightarrow M\buildo p\over\longrightarrow E$,}
where $M$ is nuclear and $p:M\to E$ is an epimorphism such that
$u\circ p$ is $A$-nuclear.
\medskip
\noindent{b)} As in (5.9), there is a $A$-nuclear decomposition
$u(x)=\sum\lambda_j\,\xi_j(x)y_j$ where $(y_j)$ converges to $0$ in~$F$.
Since $f$ is an epimorphism, we can find a sequence $(x_j)$ converging to
$0$ in $E$ such that $f(x_j)=y_j$. Hence we have $u=f\circ v$ where
$v(x)=\sum\lambda_j\,\xi_j(x)x_j$ is a $A$-nuclear endomorphism of $E$,
and the cokernel of $f-u$ is the image by $f$ of the cokernel of
$\Id_E-v$.
\medskip
\noindent{c)} By a), b) we may suppose that $F=E=A\hotimes M$, $f=\Id_E$
and that $u$ is $A$-nuclear. Let $B$ be the Banach algebra
$B=\wh A_{p_t}$. Then \hbox{$B\hotimes_A E=B\hotimes M$} is a Fr\'echet
$B$-module and $\Id_B\hotimes_A u$ is $B$-nuclear. By Lemma~5.42,
\hbox{$\Id_B\hotimes_A\Id_E-\Id_B\hotimes_A u$} has a finitely generated
cokernel over $B$. Now, there is an obvious morphism $B\to A_t$,
hence by taking the tensor product with $A_t\hotimes_B\bu$
we get
$$A_t\hotimes_B(B\hotimes_A E)=A_t\hotimes_B(B\hotimes M)=
A_t\hotimes M=A_t\hotimes_A E=E_t$$
and we see that
$$\Id_{E_t}-u_t=\Id_{A_t}\hotimes_A\Id_E-\Id_{A_t}\hotimes_A u$$
has a finitely generated cokernel over $A_t$.\qed
\endproof

\subsection{5.E}{Proof of the Direct Image Theorem}
We first prove a functional analytic version of the result, which appears
as a relative version of the Cartan-Serre/Schwartz theorem (Th.~1.9).

\claim{(5.44) Theorem} Let $A$ be a fully nuclear algebra, $E^\bu$ 
and $F^\bu$ complexes of fully $A$-transverse nuclear $A$-modules. Let
$f^\bu:E^\bu\to F^\bu$ be a morphism of complexes such that each $f^q$
is $A$-subnuclear. Suppose that $E^\bu$ and $F^\bu$ are bounded on the
right and that $H^q(f^\bu)$ is an isomorphism for each $q$. Then for
every $t<1$, there is a complex $L^\bu$ of finitely generated free
$A_t$-modules and a complex morphism $h^\bu:L^\bu\to E_t^\bu$ which induces
an isomorphism on cohomology.
\endclaim

\proof{Proof} a) We first show the following statement:
\smallskip
\item{}{\it Suppose that $E^\bu_t$ and $F^\bu_t$ are acyclic in degrees $>q$.
Then for every $t'<t$, the cohomology space $H^q(E^\bu_{t'})\simeq
H^q(F^\bu_{t'})$ is a finitely generated \hbox{$A_{t'}$-module}.}
\smallskip
Indeed, the exact sequences $0\to Z^k(E^\bu_t)\to E^k_t\to Z^{k+1}(E^\bu_t)
\to 0$ show by backward induction on $k$ that $Z^k(E^\bu_t)$ is fully
$A_t$-transverse for $k\ge q$. The same is true for $Z^k(F^\bu_t)$.
Then $f^q_t$ is a $A_t$-subnuclear map from $Z^q(E^\bu_t)$ into
$F^q_t$, and its image is contained in the closed subspace $Z^q(F^\bu_t)$.
By Prop.~5.38, for all $t''<t$, the map $f^q_{t''}=\Id_{A_{t''}}
\hotimes_{A_t}f^q_t$ is a $A_{t''}$-subnuclear map $Z^q(E^\bu_{t''})\to
Z^q(F^\bu_{t''})$. By Prop.~5.34~d), $H^\bu(f^\bu_{t''})$ is an
isomorphism in all degrees, hence
$$d^q_{t''}\oplus f^q_{t''}:F^{q-1}_{t''}\oplus Z^q(E^\bu_{t''})\longrightarrow
Z^q(F^\bu_{t''})$$
is surjective. By the subnuclear perturbation theorem, the map
$$d^q_{t'}\oplus 0=\Id_{A_{t'}}\hotimes_{A_{t''}}\big(
(d^q_{t''}\oplus f^q_{t''})-(0\oplus f^q_{t''})\big)$$
has a finitely generated $A_{t'}$-cokernel for $t'<t''<t$, as desired.
\medskip
\noindent{b)} Let $N$ be an index such that $E^k=F^k=0$ for $k>N$. Fix a
sequence $t<\ldots<t_q<t_{q+1}<\ldots<t_N<1$. To prove the theorem, we
construct by backward induction on $q$ a finitely generated free module
$L^q$ over $A_{t_q}$ and morphisms $d^q:L^q\to L^{q+1}_{t_q}$,
$h^q:L^q\to E^q_{t_q}$ such that
{\parindent=6.24mm
\smallskip
\item{\swapleft{4.5pt}{i)}}{\it $L_{\gge q,\,t_q}^\bu:0\to L^q\to
L^{q+1}_{t_q}\to\cdots\to L^N_{t_q}\to 0$ is a complex and\Fbreak
$h^\bu_{\gge q,\,t_q}:L_{\gge q,\,t_q}^\bu\to E^\bu_{t_q}$
is a complex morphism.}
\smallskip
\item{\swapleft{4.5pt}{ii)}}{\it The mapping cylinder $M^\bu_q=
C(h^\bu_{\gge q,\,t_q})$ defined by\Fbreak
$M^k_q=\bigoplus_{k\in\bZ}\big(L^k_{\gge q,\,t_q}\oplus E^{k-1}_{t_q}\big)$
is acyclic in degrees $k>q$.}
\smallskip
}
\noindent
Suppose that $L^k$ has been constructed for $k\ge q$. Consider the
mapping cylinder $N^\bu_q=C(f^\bu_{t_q}\circ h^\bu_{\gge q,\,t_q})$
and the complex morphism
$$M^\bu_q\longrightarrow N^\bu_q,~~~~
L^k_{\gge q,\,t_q}\oplus E^{k-1}_{t_q}\longrightarrow
L^k_{\gge q,\,t_q}\oplus F^{k-1}_{t_q}$$
given by $\Id\oplus f^{k-1}_{t_q}$. This morphism is
$A_{t_q}$-subnuclear in each degree and induces an isomorphism
in cohomology (compare the cohomology of the short exact sequences
associated to each mapping cylinder, with the obvious morphism
between them). Moreover, $M^\bu_q$ and $N^\bu_q$ are acyclic in degrees
\hbox{$k>q$}. By step a), the cohomology space $H^q(M^\bu_{q,\,t_{q-1}})$
is a finitely generated \hbox{$A_{t_{q-1}}$-module}.
Therefore, we can find a finitely generated free $A_{t_{q-1}}$-module
$L^{q-1}$ and a morphism
$$d^{q-1}\oplus h^{q-1}:L^{q-1}\to M^q_{q,\,t_{q-1}}=L^q_{t_{q-1}}\oplus
E^{q-1}_{t_{q-1}}$$
such that the image is contained in $Z^q(M^\bu_{q,\,t_{q-1}})$ and
generates the cohomology space $H^q(M^\bu_{q,\,t_{q-1}})$. As
$M^{q-1}_{q,\,t_{q-1}}=E^{q-2}_{t_{q-1}}$, this means that
$M^\bu_{q-1}$ is also acyclic in degree $q$. Thus $L^{q-1}$, together
with the maps $(d^{q-1},h^{q-1})$ satisfies the induction hypotheses
for $q-1$, and $L^\bu_t$ together with the induced map
$h^\bu_t:L^\bu_t\to E^\bu_t$ is the required morphism of complexes.\qed
\endproof

\proof{Proof of Theorem 5.1} Let $X$, $Y$ be complex analytic
schemes, let \hbox{$F:X\to Y$} be a proper analytic morphism and let
$\cS$ be a coherent sheaf over $X$. Fix a point $y_0\in Y$, a neighborhood
of $y_0$ which is isomorphic to a closed analytic subscheme of a Stein
open set $W\subset\bC^n$ and a polydisk $D^0=D(y_0,R_0)\compact W$.
The compact set $K=F^{-1}(\ol D^0\cap Y)$ can be covered
by finitely many open subsets $U_\alpha^0\compact X$ which possess
embeddings as closed analytic subschemes of Stein open sets
$\Omega_\alpha^0\subset\bC^{N_\alpha}$. Let
$\Omega'_\alpha\compact\Omega_\alpha\compact\Omega_\alpha^0$ be
Stein open subsets such that $U_\alpha=U^0_\alpha\cap\Omega_\alpha$
and $U'_\alpha=U^0_\alpha\cap\Omega'_\alpha$ still cover $K$.
Let $i_\alpha:U^0_\alpha\to\Omega^0_\alpha$ and $j:Y\cap D^0\to D^0$ be the
embeddings and $\cS_\alpha=\big(i_\alpha\times(j\circ F)\big)_\star\cS$ the
image sheaf of $\cS$ on $\Omega^0_\alpha\times D^0$. Let $D\compact D^0$
be a concentric polydisk. Then $\cS\big(U_\alpha\cap F^{-1}(D)\big)=
\cS_\alpha(\Omega_\alpha\times D)$ is a fully transverse $\cO(D)$-module
by Ex.~5.41, and so is $\cS\big(U'_\alpha\cap F^{-1}(D)\big)=
\cS_\alpha(\Omega'_\alpha\times D)$. Moreover, the restriction map
$$\cS\big(U_\alpha\cap F^{-1}(D)\big)\longrightarrow
\cS\big(U'_\alpha\cap F^{-1}(D)\big)$$
is $\cO(D)$-subnuclear by Prop.~5.37 applied to $\cF=\cS_\alpha$. For
every Stein open set $V\subset D$, Prop.~5.33  shows that
$$\cO(V)\hotimes_{\cO(D)}\cS\big(U_\alpha\cap F^{-1}(D)\big)
=\cS\big(U_\alpha\cap F^{-1}(V)\big).$$

Denote by $\cU\cap F^{-1}(D)$ the collection $\big(U_\alpha\cap F^{-1}(D)
\big)$ and use a similar notation with $\cU'=(U'_\alpha)$.
As $\cU\cap F^{-1}(D)$, $\cU'\cap F^{-1}(D)$ are Stein coverings of
$F^{-1}(D)$, the Leray theorem applied to the alternate \v Cech complex of
$\cS$ over $\cU\cap F^{-1}(D)$ and $\cU'\cap F^{-1}(D)$ gives an isomorphism
$$H^\bu\big(AC^\bu(\cU\cap F^{-1}(D),\cS)\big)=
H^\bu\big(AC^\bu(\cU'\cap F^{-1}(D),\cS)\big)=H^\bu\big(F^{-1}(D),\cS\big).$$
By the above discussion, $AC^\bu(\cU\cap F^{-1}(D),\cS)$ and
$AC^\bu(\cU'\cap F^{-1}(D),\cS)$ are finite complexes of fully
transverse nuclear $\cO(D)$-modules, the restriction map
$$AC^\bu(\cU\cap F^{-1}(D),\cS)\longrightarrow AC^\bu(\cU'\cap F^{-1}(D),\cS)$$
is $\cO(D)$-subnuclear and induces an isomorphism on cohomology groups.
Set \hbox{$D=D(y_0,R)\kern-0.85pt$} and $D_t=D(y_0,tR)$. Theorem~5.44
shows that for every $t<1$
there is a complex of finitely generated free $\cO$-modules $\cL^\bu$ and a
\hbox{$\cO(D_t)$-linear} morphism of complexes
$$\cL^\bu(D_t)\to AC^\bu(\cU\cap F^{-1}(D_t),\cS)$$ which
induces an isomorphism on cohomology. Let $V\subset D_t$ be an arbitrary
Stein open set. By Prop.~5.34~d) applied with $M=\cO(V)$, we conclude
that $\cL^\bu(V)\to AC^\bu(\cU\cap F^{-1}(V),\cS)$ induces an
isomorphism on cohomology. If we take the direct limit as $V$ runs
over all Stein neighborhoods of a point $y\in Y\cap D_t$, we see that
$\cH^q(\cL^\bu)\simeq R^qF_\star\cS$ over $Y\cap D_t$, hence
$R^qF_\star\cS$ is $\cO_Y$-coherent near $y_0$.\qed
\endproof

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\endeightpoint
\vfill\eject

\end

